NCERT Solutions for Class 12 Maths Chapter 6 - Application of Derivatives Free PDF Download

Exercise- 6.1

1. Find the rate of change of the area of a circle with respect to its radius $r$ when

a. $r = 3\;{\text{cm}}$ (b) $r = 4\;{\text{cm}}$

Ans: We know that $A = \pi {r^2}$Differentiate w.r.t \[r\]

$\therefore \dfrac{{dA}}{{dr}} = \dfrac{d}{{dr}}\left( {\pi {r^2}} \right) = 2\pi r$

(a) When $r = 3\;{\text{cm}}$,

$\dfrac{{dA}}{{dr}} = 2\pi (3) = 6\pi$

The area is changing at $6\;{\text{c}}{{\text{m}}^2}/{\text{s}}$ when radius is $3\;{\text{cm}}$.

b. When $r = 4\;{\text{cm}}$,

$\dfrac{{dA}}{{dr}} = 2\pi (4) = 8x$

The area is changing at $8\;{\text{c}}{{\text{m}}^2}/{\text{s}}$ when radius is $4\;{\text{cm}}$.

2. The volume of a cube is increasing at the rate of $8\;{\text{c}}{{\text{m}}^3}/{\text{s}}$ How fast is the surface area increasing when the length of its edge is $12\;{\text{cm}}$ ?

Ans: Let the side length, volume and surface area respectively be equal to \[x,v\] and $S$. $V = {x^3}$

$S = 6{x^2}$

$\dfrac{{dV}}{{dt}} = 8\;{\text{c}}{{\text{m}}^3}/{\text{s}}$

$\therefore 8 = \dfrac{{dV}}{{dt}} = \dfrac{d}{{dt}}\left( {{x^3}} \right)$

$= \dfrac{d}{{dx}}\left( {{x^3}} \right)\dfrac{{dx}}{{dt}} = 3{x^2}\dfrac{{dx}}{{dt}}$

$\therefore 8 = \dfrac{{dV}}{{dt}} = \dfrac{d}{{dt}}\left( {{x^3}} \right) = \dfrac{d}{{dx}}\left( {{x^3}} \right)\dfrac{{dx}}{{dt}} = 3{x^2}\dfrac{{dx}}{{dt}} \Rightarrow \dfrac{{dx}}{{dt}} = \dfrac{8}{{3{x^2}}}$

$\dfrac{{dS}}{{dt}} = \dfrac{d}{{dt}}\left( {6{x^2}} \right)$

$= \dfrac{d}{{dx}}\left( {6{x^2}} \right)\dfrac{{dx}}{{dt}}$

$\dfrac{{dS}}{{dt}} = \dfrac{d}{{dt}}\left( {6{x^2}} \right) = \dfrac{d}{{dx}}\left( {6{x^2}} \right)\dfrac{{dx}}{{dt}} = 12x\dfrac{{dx}}{{dt}} = 12x\left( {\dfrac{8}{{3{x^2}}}} \right) = \dfrac{{32}}{x}$

So, when $x = 12\;{\text{cm}},$

$\dfrac{{dS}}{{dt}} = \dfrac{{32}}{{12}}\;{\text{c}}{{\text{m}}^2}/{\text{s}}$

$x = 12\;{\text{cm}},\dfrac{{dS}}{{dt}} = \dfrac{{32}}{{12}}\;{\text{c}}{{\text{m}}^2}/{\text{s}} = \dfrac{8}{3}\;{\text{c}}{{\text{m}}^2}/{\text{s}}.$

3. The radius of a circle is increasing uniformly at the rate of $3\;{\text{cm}}/{\text{s}}$. Find the rate at which the area of the circle is increasing when the radius is $10\;{\text{cm}}/{\text{s}}$

Ans: We know that $A = \pi {r^2}$ $\therefore \dfrac{{dA}}{{dt}} = \dfrac{d}{{dr}}\left( {\pi {r^2}} \right)\dfrac{{dr}}{{dt}} = 2\pi r\dfrac{{dr}}{{dt}}$

$\dfrac{{dr}}{{dt}} = 3\;{\text{cm}}/{\text{s}}$

$\therefore \dfrac{{dA}}{{dt}} = 2\pi r(3) = 6\pi r$

So, when $r = 10\;{\text{cm}}$,

$\dfrac{{dA}}{{dt}} = 6\pi (10) = 60\pi {\text{c}}{{\text{m}}^2}/{\text{s}}$

4. An edge of a variable cube is increasing at the rate of $3\;{\text{cm}}/{\text{s}}$. How fast is the volume of the cube increasing when the edge is $10\;{\text{cm}}$ long? 

Ans: Let the length and the volume of the cube respectively be $x$ and $V$.

 $V = {x^3}$

$\therefore \dfrac{{dV}}{{dt}} = \dfrac{d}{{dt}}\left( {{x^3}} \right) = \dfrac{d}{{dx}}\left( {{x^3}} \right)\dfrac{{dx}}{{dt}}$

$= 3{x^2}\dfrac{{dx}}{{dt}}$

$\dfrac{{dx}}{{dt}} = 3\;{\text{cm}}/{\text{s}}$

$\therefore \dfrac{{dV}}{{dt}} = 3{x^2}(3) = 9{x^2}$

So, when $x = 10\;{\text{cm}}$,

$\dfrac{{dV}}{{dt}} = 9{(10)^2} = 900\;{\text{c}}{{\text{m}}^3}/{\text{s}}$

5. A stone is dropped into a quiet lake and waves move in circles at the speed of $5\;{\text{cm}}/{\text{s}}.$ At the instant when the radius of the circular wave is $8\;{\text{cm}}$, how fast is the enclosed area increasing?

Ans: We know that $A = \pi {r^2}$ 

$\dfrac{{dA}}{{dt}} = \dfrac{d}{{dt}}\left( {\pi {r^2}} \right) = \dfrac{d}{{dr}}\left( {\pi {r^2}} \right)\dfrac{{dr}}{{dt}} = 2\pi r\dfrac{{dr}}{{dt}}$

$\dfrac{{dr}}{{dt}} = 5\;{\text{cm}}/{\text{s}}$

So, when $r = 8\;{\text{cm}}$, 

$\dfrac{{dA}}{{dt}} = 2\pi (8)(5)$

$= 80\pi {\text{c}}{{\text{m}}^2}/{\text{s}}.$

 6. The radius of a circle is increasing at the rate of $0.7\;{\text{cm}}/{\text{s}}$. What is the rate of increase of its circumference?

Ans: We know that $C = 2\pi r$.

 $\therefore \dfrac{{dC}}{{dt}} = \dfrac{{dC}}{{dr}}\dfrac{{dr}}{{dt}} = \dfrac{d}{{dr}}(2\pi r)\dfrac{{dr}}{{dt}} = 2\pi \dfrac{{dr}}{{dt}}$

$\dfrac{{dr}}{{dt}} = 0.7\;{\text{cm}}/{\text{s}}$

$\therefore \dfrac{{dC}}{{dt}} = 2\pi (0.7) = 1.4\pi {\text{cm}}/{\text{s}}$

7. The length $x$ of a rectangle is decreasing at the rate of $5\;{\text{cm}}/$ minute and the width $y$ is increasing at the rate of $4\;{\text{cm}}/$ minute. When $x = 8\;{\text{cm}}$ and $y = 6\;{\text{cm}}$, find the rates of change of $(a)$ the perimeter, and $(b)$ the area of the rectangle. 

Ans: It is given that $\dfrac{{dx}}{{dt}} =  - 5\;{\text{cm}}/{\text{min}},$$\dfrac{{dx}}{{dt}} = 4\;{\text{cm}}/{\text{min}},$ $x = 8\;{\text{cm}}$and $y = 6\;{\text{cm}}$,

1. The perimeter of a rectangle is given by $P = 2(x + y)$

$\therefore \dfrac{{dP}}{{dt}} = 2\left( {\dfrac{{dx}}{{dt}} + \dfrac{{dy}}{{dt}}} \right) = 2( - 5 + 4) =  - 2\;{\text{cm}}/{\text{min}}$

2. The area of a rectangle is given by $A = xy$ 

3. $\therefore \dfrac{{dA}}{{dt}} = \dfrac{{dx}}{{dt}}y + x\dfrac{{dy}}{{dt}} =  - 5y + 4x$

When $x = 8\;{\text{cm}}$ and $y = 6\;{\text{cm}},$

$\dfrac{{dA}}{{dt}} = ( - 5 \times 6 + 4 \times 8){\text{c}}{{\text{m}}^2}/{\text{min}} = 2\;{\text{c}}{{\text{m}}^2}/{\text{min}}$

8. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15\;{\text{cm}}$.

Ans: We know that $V = \dfrac{4}{3}\pi {r^3}$ 

$\therefore \dfrac{{dV}}{{dt}} = \dfrac{{dV}}{{dr}}\dfrac{{dr}}{{dt}} = \dfrac{d}{{dr}}\left( {\dfrac{4}{3}\pi {r^2}} \right)\dfrac{{dr}}{{dt}} = 4\pi {r^2}\dfrac{{dr}}{{dt}}$

$\dfrac{{dV}}{{dt}} = 900\;{\text{c}}{{\text{m}}^2}/{\text{s}}$

$\therefore 900 = 4\pi {r^2}\dfrac{{dr}}{{dt}}$

$\Rightarrow \dfrac{{dr}}{{dt}} = \dfrac{{900}}{{4\pi {r^2}}} = \dfrac{{225}}{{\pi {r^2}}}$

So, when radius $= 15\;{\text{cm}}$,

 $\dfrac{{dr}}{{dt}} = \dfrac{{225}}{{\pi {{(15)}^2}}} = \dfrac{1}{\pi }{\text{cm}}/{\text{s}}$

9. A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the latter is $10\;{\text{cm}}$. 

Ans: We know that $V = \dfrac{4}{3}\pi {r^2}$

$\therefore \dfrac{{dV}}{{dr}} = \dfrac{d}{{dr}}\left( {\dfrac{4}{3}\pi {r^3}} \right) = \dfrac{4}{3}\pi \left( {3{r^2}} \right) = 4\pi {r^2}$

So, when radius $= 10\;{\text{cm}},\dfrac{{dV}}{{dr}} = 4\pi {(10)^2} = 400\pi $

Thus, the volume of the balloon is increasing at the rate of $400\pi {\text{c}}{{\text{m}}^3}/{\text{s}}$.

10. A ladder $5\;{\text{m}}$ long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of $2\;{\text{cm}}/{\text{s}}$. How fast is its height on the wall decreasing when the foot of the ladder is $4\;{\text{m}}$ away from the wall? 

Ans: Let the heigt of the wall at which the ladder is touching it be $y\;{\text{m}}$ and the distance of its foot from the wall on the ground be $x\;{\text{m}}$.

$\therefore {x^2} + {y^2} = {5^2} = 25$

   $\Rightarrow y = \sqrt {25 - {x^2}}$

$\therefore \dfrac{{dy}}{{dt}} = \dfrac{d}{{dt}}\left( {\sqrt {25 - {x^2}} } \right) = \dfrac{d}{{dx}}\left( {\sqrt {25 - {x^2}} } \right)\dfrac{{dx}}{{dt}} = \dfrac{{ - x}}{{\sqrt {25 - {x^2}} }}\dfrac{{dx}}{{dt}}$

$\dfrac{{dx}}{{dt}} = 2\;{\text{cm}}/{\text{s}}$

$\therefore \dfrac{{dy}}{{dt}} = \dfrac{{ - 2x}}{{\sqrt {25 - {x^2}} }}$

So, when $x = 4m$,

 $\dfrac{{dy}}{{dt}} = \dfrac{{ - 2 \times 4}}{{\sqrt {25 - 16} }} =  - \dfrac{8}{3}$

11. A particle is moving along the curve $6y = {x^3} + 2.$ Find the points on the curve at which the $Y$ coordinate is changing 8 times as fast as the ${\text{X}}$ coordinate. 

Ans: The equation of the curve is $6y = {x^3} + 2$. Differentiating with respect to time, we have, $6\dfrac{{dy}}{{dt}} = 3{x^2}\dfrac{{dx}}{{dt}}$

$\Rightarrow 2\dfrac{{dy}}{{dt}} = {x^2}\dfrac{{dx}}{{dt}}$

According to the question, $\left( {\dfrac{{dy}}{{dt}} = 8\dfrac{{dx}}{{dt}}} \right)$

$\therefore 2\left( {8\dfrac{{dx}}{{dt}}} \right) = {x^2}\dfrac{{dx}}{{dt}} \Rightarrow 16\dfrac{{dx}}{{dt}} = {x^2}\dfrac{{dx}}{{dt}}$

$\Rightarrow \left( {{x^2} - 16} \right)\dfrac{{dx}}{{dt}} = 0$

$\Rightarrow {x^2} = 16$

$\Rightarrow x =  \pm 4$

When $x = 4,$

$y = \dfrac{{{4^3} + 2}}{6} = \dfrac{{66}}{6} = 11$

When $x =  - 4,$

$y = \dfrac{{\left( { - {4^3}} \right) + 2}}{6} =  - \dfrac{{62}}{6} =  - \dfrac{{31}}{3}$

Thus, the points on the curve are $(4,11)$ and $\left( { - 4,\dfrac{{ - 31}}{3}} \right)$

12. The radius of an air bubble is increasing at the rate of $\dfrac{1}{2}\;{\text{cm}}/{\text{s}}$. At what rate is the volume of the bubble increasing when the radius is $1\;{\text{cm}}$ ? 

Ans: Assuming that the air bubble is a sphere, $V = \dfrac{4}{3}\pi {r^2}$

$\therefore \dfrac{{dV}}{{dt}} = \dfrac{d}{{dt}}\left( {\dfrac{{4\pi }}{3}{r^3}} \right) = \dfrac{d}{{dr}}\left( {\dfrac{{4\pi }}{3}{r^3}} \right)\dfrac{{dr}}{{dt}} = 4\pi {r^2}\dfrac{{dr}}{{dt}}$

$\dfrac{{dr}}{{dt}} = \dfrac{1}{2}\;{\text{cm}}/{\text{s}}$

So, when $r = 1\;{\text{cm}},$

$\dfrac{{dV}}{{dt}} = 4\pi {(1)^2}\left( {\dfrac{1}{2}} \right) = 2\pi {\text{c}}{{\text{m}}^3}/{\text{s}}$

13. A balloon, which always remains spherical, has a variable diameter $\dfrac{3}{2}(2x + 1)$. Find the rate of change of its volume with respect to $x$. 

Ans: We know that $V = \dfrac{4}{3}\pi {r^3}$

 $d = \dfrac{3}{2}(2x + 1)$

$\Rightarrow r = \dfrac{3}{4}(2x + 1)$

$\therefore V = \dfrac{4}{3}\pi {\left( {\dfrac{3}{4}} \right)^3}{(2x + 1)^3} = \dfrac{9}{{16}}\pi {(2x + 1)^3}$

$\therefore \dfrac{{dV}}{{dx}} = \dfrac{9}{{16}}\pi \dfrac{d}{{dx}}{(2x + 1)^3} = \dfrac{{27}}{8}\pi {(2x + 1)^3}.$

14. Sand is pouring from a pipe at the rate of $12\;{\text{c}}{{\text{m}}^3}/{\text{s}}$. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is $4\;{\text{cm}}$ ?

Ans: We know that $V = \dfrac{1}{3}\pi {r^2}h$ 

$h = \dfrac{1}{6}r$

$\Rightarrow r = 6h$

$\therefore V = \dfrac{1}{3}\pi {(6h)^2}h = 12\pi {h^3}$

$\therefore \dfrac{{dV}}{{dt}} = 12\pi \dfrac{d}{{dt}}\left( {{h^3}} \right)\dfrac{{dh}}{{dt}}$

$= 12\pi \left( {3{h^2}} \right)\dfrac{{dh}}{{dt}} = 36\pi {h^2}\dfrac{{dh}}{{dt}}$

$\dfrac{{dV}}{{dt}} = 12\;{\text{c}}{{\text{m}}^2}/{\text{s}}$

So, when $h = 4\;{\text{cm}}$,

$12 = 36\pi {(4)^2}\dfrac{{dh}}{{dt}}$

$\Rightarrow \dfrac{{dh}}{{dt}} = \dfrac{{12}}{{36\pi (16)}}$

$= \dfrac{1}{{48\pi }}{\text{cm}}/{\text{s}}$

15. The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007{x^3} - 0.003{x^2} + 15x + 4000$. Find the marginal cost when 17 units are produced.

Ans: Marginal cost is the rate of change of the total cost with respect to the output.

$\therefore $ Marginal cost $MC = \dfrac{{dC}}{{dx}} = 0.007\left( {3{x^2}} \right) - 0.003(2x) + 15 = 0.021{x^2} - 0.006x + 15$

When $x = 17,MC = 0.021\left( {{{17}^2}} \right) - 0.006(17) + 15$

$= 0.021(289) - 0.006(17) + 15$

$= 6.069 - 0.102 + 15$

$= 20.967$

So, when 17 units are produced, the marginal cost is Rs.\[20.967\].

16. The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 13{x^2} + 26x + 15$. Find the marginal revenue when $x = 7$.

Ans: Marginal revenue is the rate of change of the total revenue with respect to the number of units sold.

$\therefore $ Marginal Revenue $MR = \dfrac{{dR}}{{dx}} = 13(2x) + 26 = 26x + 26$

When $x = 7,MR = 26(7) + 26 = 182 + 26 = 208$

Thus, the marginal revenue is Rs \[208.\]

17. The rate of change of the area of a circle with respect to its radius $r$ at $r = 6\;{\text{cm}}$ is

\[{\text{(A) }}10\pi\]

\[{\text{(B) }}12\pi\]

\[{\text{(C) }}8\pi\]

\[{\text{(D) }}11\pi\]

Ans: We know that $A = \pi {r^2}$ 

$\therefore \dfrac{{dA}}{{dr}} = \dfrac{d}{{dr}}\left( {\pi {r^2}} \right) = 2\pi r$

So, when $r = 6\;{\text{cm}},$

$\dfrac{{dA}}{{dr}} = 2\pi  \times 6 = 12\pi {\text{c}}{{\text{m}}^2}/{\text{s}}$

Thus, the rate of change of the area of the circle is $12\pi {\text{c}}{{\text{m}}^2}/{\text{s}}$. 

The correct answer is option ${\mathbf{B}}$.

18. The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 3{x^2} + 36x + 5$. The marginal revenue, when $x = 15$ is

1. 116

2. 96

3. 90

4. 126 

Ans: Marginal revenue is the rate of change of the total revenue with respect to the number of units sold.

$\therefore $ Marginal Revenue $MR = \dfrac{{dR}}{{dx}} = 3(2x) + 36$

$= 6x + 36$

So, when $x = 15,$

$MR = 6(15) + 36 = 90 + 36 = 126$

Hence, the marginal revenue is Rs\[126\]. 

The correct answer is option D.

Exercise 6.2

1. Show, that the function given by $f(x) = 3x + 17$ is strictly increasing on ${\mathbf{R}}$.

Ans: Let ${x_1}$ and ${x_2}$, be any two numbers in $\mathbb{R}$.

${x_1} < {x_2} \Rightarrow 3{x_1} + 17 < 3{x_2} + 17 = f\left( {{x_1}} \right) < f\left( {{x_2}} \right)$

Thus, $f$ is strictly increasing on $\mathbb{R}$.

Alternate Method:

${f^\prime }(x) = 3 > 0$ on $\mathbb{R}$.

Thus, $f$ is strictly increasing on $\mathbb{R}$.

2. Show, that the function given by $f(x) = {e^{2x}}$ is strictly increasing on $\mathbb{R}$.

Ans: Let ${x_1}$ and ${x_2}$ be any two numbers in $\mathbb{R}$. 

${x_1} < {x_2}$

$\Rightarrow 2{x_1} < 2{x_2}$

$\Rightarrow {e^{2{x_1}}} < {e^{2{x_2}}}$

$\Rightarrow f\left( {{x_1}} \right) < f\left( {{x_2}} \right)$

Thus, $f$ is strictly increasing on $\mathbb{R}$

3. Show that the function given by $f(x) = \sin x$ is

1. Strictly increasing in $\left( {0,\dfrac{\pi }{2}} \right)$

2. Strictly decreasing $\left( {\dfrac{\pi }{2},\pi } \right)$

3. Neither increasing nor decreasing in $(0,\pi )$

Ans: $f(x) = \sin x \Rightarrow {f^\prime }(x) = \cos x$

1. $x \in \left( {0,\dfrac{\pi }{2}} \right)$

$\Rightarrow \cos x > 0 \Rightarrow {f^\prime }(x) > 0$

Thus, $f$ is strictly increasing in $\left( {0,\dfrac{\pi }{2}} \right)$.

2. $x \in \left( {\dfrac{\pi }{2},\pi } \right)$

$\Rightarrow \cos x < 0 \Rightarrow {f^\prime }(x) < 0$

Thus, $f$ is strictly decreasing in $\left( {\dfrac{\pi }{2},\pi } \right)$.

3. The results obtained in (A) and (B) are sufficient to state that $f$ is neither increasing nor decreasing in $(0,\pi )$.

4. Find the intervals in which the function $f$ given by $f(x) = 2{x^2} - 3x$ is

1. Strictly increasing

2. Strictly decreasing 

Ans: $f(x) = 2{x^2} - 3x \Rightarrow {f^\prime }(x) = 4x - 3$

$\therefore {f^\prime }(x) = 0 \Rightarrow x = \dfrac{3}{4}$

In $\left( { - \infty ,\dfrac{3}{4}} \right),$${f^\prime }(x) = 4x - 3 < 0$

Hence, $f$ is strictly decreasing in $\left( { - \infty ,\dfrac{3}{4}} \right)$

In $\left( {\dfrac{3}{4},\infty } \right),$${f^\prime }(x) = 4x - 3 > 0$

Hence, $f$ is strictly increasing in $\left( {\dfrac{3}{4},\infty } \right)$

5. Find the intervals in which the function $f$ given $f(x) = 2{x^2} - 3{x^2} - 36x + 7$ is

1. Strictly increasing 

2. Strictly decreasing 

Ans: $f(x) = 2{x^3} - 3{x^2} - 36x + 7$

${f^\prime }(x) = 6{x^2} - 6x - 36 = 6\left( {{x^2} - x - 6} \right) = 6(x + 2)(x - 3)$

$\therefore {f^\prime }(x) = 0 \Rightarrow x =  - 2,3$

In $( - \infty , - 2)$ and $(3,\infty ),{f^\prime }(x) > 0$

In $( - 2,3),{f^\prime }(x) < 0$

Hence, $f$ is strictly increasing in $( - \infty , - 2)$ and $(3,\infty )$ and strictly decreasing in $( - 2,3)$.

6. Find the intervals in which the following functions are strictly increasing or decreasing.

1. ${x^2} + 2x - 5$

2. $10 - 6x - 2{x^2}$

3. $- 2{x^3} - 9{x^2} - 12x + 1$

4. $6 - 9x - {x^2}$

5. ${(x + 1)^3}{(x - 3)^3}$

Ans: 

1. $f(x) = {x^2} + 2x - 5$

$\Rightarrow {f^\prime }(x) = 2x + 2$

$\Rightarrow {f^\prime }(x) = 0 \Rightarrow x =  - 1$

$x =  - 1$ divides the number line into intervals $( - \infty , - 1)$ and $( - 1,\infty ).$

In $( - \infty , - 1),{f^\prime }(x) = 2x + 2 < 0$

$\therefore f$ is strictly decreasing in $( - \infty , - 1)$

In $( - 1,\infty ),{f^\prime }(x) = 2x + 2 > 0,$

$\therefore {f^\prime }(x) = 2x + 2 > 0$

$\therefore f$ is strictly decreasing in $( - 1,\infty )$

2. $f(x) = 10 - 6x - 2{x^2}$

$\Rightarrow {f^\prime }(x) =  - 6 - 4x$

$\Rightarrow {f^\prime }(x) = 0 \Rightarrow x =  - \dfrac{3}{2}$

$x =  - \dfrac{3}{2}$ divides the number line into two intervals $\left( { - \infty , - \dfrac{3}{2}} \right)$ and $\left( { - \dfrac{3}{2},\infty } \right)$

In $\left( { - \infty , - \dfrac{3}{2}} \right),$${f^\prime }(x) =  - 6 - 4x < 0$

$\therefore f$ is strictly increasing for $x <  - \dfrac{3}{2}$

In $\left( { - \dfrac{3}{2},\infty } \right),{f^\prime }(x) =  - 6 - 4x > 0$.

$\therefore f$ is strictly increasing for $x >  - \dfrac{3}{2}$

3. $f(x) =  - 2{x^3} - 9{x^2} - 12x + 1$

$\therefore {f^\prime }(x) =  - 6{x^2} - 18x - 12 =  - 6\left( {{x^2} + 3x + 2} \right) =  - 6(x + 1)(x + 2)$

${f^\prime }(x) = 0 \Rightarrow x =  - 1,2$

$x =  - 1$ and $x =  - 2$ divide the number line into intervals $( - \infty , - 2),( - 2, - 1)$ and $( - 1,\infty )$

In $( - \infty , - 2)$ and $( - 1,\infty ),{f^\prime }(x) =  - 6(x + 1)(x + 2) < 0$

$\therefore f$ is strictly decreasing for $x <  - 2$ and $x >  - 1$.

In $( - 2, - 1),{f^\prime }(x) =  - 6(x + 1)(x + 2) > 0$

$\therefore f$ is strictly increasing for $ - 2 < x <  - 1$

4. $f(x) = 6 - 9x - {x^2}$

$\Rightarrow {f^\prime }(x) =  - 9 - 2x$

${f^\prime }(x) = 0 \Rightarrow x = \dfrac{{ - 9}}{2}$

$\operatorname{In} \left( { - \infty , - \dfrac{9}{2}} \right),{f^\prime }(x) > 0$

$\therefore f$ is strictly increasing for $x <  - \dfrac{9}{2}$

$\operatorname{In} \left( { - \dfrac{9}{2},\infty } \right),{f^\prime }(x) < 0$

$\therefore f$ is strictly decreasing for $x >  - \dfrac{9}{2}$

5. $f(x) = {(x + 1)^3}{(x - 3)^3}$

${f^\prime }(x) = 3{(x + 1)^2}{(x - 3)^3} + 3{(x - 3)^2}{(x + 1)^3}$

$= 3{(x + 1)^2}{(x - 3)^2}[x - 3 + x + 1]$

$= 3{(x + 1)^2}{(x - 3)^2}(2x - 2)$

$= 6{(x + 1)^2}{(x - 3)^2}(x - 1)$

${f^\prime }(x) = 0$

$\Rightarrow x =  - 1,3,1$

$x =  - 1,3,1$ divides the number line into four intervals $( - \infty , - 1),( - 1,1),(1,3)$ and $(3,\infty )$

In $( - \infty , - 1)$ and $( - 1,1),{f^\prime }(x) = 6{(x + 1)^2}{(x - 3)^2}(x - 1) < 0$

$\therefore f$ is strictly decreasing in $( - \infty , - 1)$ and $( - 1,1)$

In $(1,3)$ and $(3,\infty ),{f^\prime }(x) = 6{(x + 1)^2}{(x - 3)^2}(x - 1) > 0$

$\therefore f$ is strictly increasing in $(1,3)$ and $(3,\infty )$

7. Show that $y = \log (1 + x) - \dfrac{{2x}}{{2 + x}},x >  - 1$, is an increasing function throughout its domain. 

Ans: $y = \log (1 + x) - \dfrac{{2x}}{{2 + x}}$

$\therefore \dfrac{{dy}}{{dx}} = \dfrac{1}{{1 + x}} - \dfrac{{(2 + x)(2) - 2x(1)}}{{{{(2 + x)}^2}}} = \dfrac{1}{{1 + x}} - \dfrac{4}{{{{(2 + x)}^2}}} = \dfrac{{{x^2}}}{{(1 + x){{(2 + x)}^2}}}$

$\dfrac{{dy}}{{dx}} = 0$

$\Rightarrow \dfrac{{{x^2}}}{{{{(2 + x)}^2}}} = 0$

$\Rightarrow {x^2} = 0$

$\Rightarrow x = 0$

Because $x >  - 1,x = 0$ divides domain $( - 1,\infty )$ in two intervals $ - 1 < x < 0$ and $x > 0$. When $ - 1 < x < 0$, $x < 0 \Rightarrow {x^2} > 0$

$x >  - 1 \Rightarrow (2 + x) > 0$

$\Rightarrow {(2 + x)^2} > 0$

$\therefore {y^\prime } = \dfrac{{{x^2}}}{{{{(2 + x)}^2}}} > 0$

When $x > 0$, 

$\Rightarrow {x^2} > 0,{(2 + x)^2} > 0$

$\therefore {y^\prime } = \dfrac{{{x^2}}}{{{{(2 + x)}^2}}} > 0$

Hence, $f$ is increasing throughout the domain.

8. Find the values of $x$ for which $y = {[x(x - 2)]^2}$ is an increasing function.

Ans: $y = {[x(x - 2)]^2} = {\left[ {{x^2} - 2x} \right]^2}$

$\therefore \dfrac{{dy}}{{dx}} = {y^\prime } = 2\left( {{x^2} - 2x} \right)(2x - 2) = 4x(x - 2)(x - 1)$

$\therefore \dfrac{{dy}}{{dx}} = 0$

$\Rightarrow x = 0,x = 2,x = 1$

$x = 0,x = 1$ and $x = 2$ divide the number line into intervals $( - \infty ,0),(0,1),(1,2)$ and $(2,\infty )$ In $( - \infty ,0)$ and $(1,2),\dfrac{{dy}}{{dx}} < 0$

$\therefore y$ is strictly decreasing in intervals $( - \infty ,0)$ and $(1,2)$ In intervals $(0,1)$ and $(2,\infty ),\dfrac{{dy}}{{dx}} > 0$

$\therefore y$ is strictly increasing in intervals $(0,1)$ and $(2,\infty )$

9. Prove that $y = \dfrac{{4\sin \theta }}{{(2 + \cos \theta )}} - \theta $ is an increasing function of $\theta $ in $\left[ {0,\dfrac{\pi }{2}} \right]$. 

Ans: $y = \dfrac{{4\sin \theta }}{{(2 + \cos \theta )}} - \theta $

$\therefore \dfrac{{dy}}{{d\theta }} = \dfrac{{(2 + \cos \theta )(4\cos \theta ) - 4\sin \theta ( - \sin \theta )}}{{{{(2 + \cos \theta )}^2}}} - 1$

$= \dfrac{{8\cos \theta  + 4{{\cos }^2}\theta  + 4{{\sin }^2}\theta }}{{{{(2 + \cos \theta )}^2}}} - 1$

$= \dfrac{{8\cos \theta  + 4}}{{{{(2 + \cos \theta )}^2}}} - 1$

$\dfrac{{dy}}{{d\theta }} = 0$

$\Rightarrow \dfrac{{8\cos \theta  + 4}}{{{{(2 + \cos \theta )}^2}}} = 1$

$\Rightarrow 8\cos \theta  + 4 = 4 + {\cos ^2}\theta  + 4\cos \theta $

$\Rightarrow {\cos ^2}\theta  - 4\cos \theta  = 0$

$\Rightarrow \cos \theta (\cos \theta  - 4) = 0$

$\Rightarrow \cos \theta  = 0$ or $\cos \theta  = 4$

Because $\cos \theta  \ne 4,\cos \theta  = 0$

$\cos \theta  = 0 \Rightarrow \theta  = \dfrac{\pi }{2}$

$\dfrac{{dy}}{{d\theta }} = \dfrac{{8\cos \theta  + 4 - \left( {4 + {{\cos }^2}\theta  + 4\cos \theta } \right)}}{{{{(2 + \cos \theta )}^2}}} = \dfrac{{4\cos \theta  - {{\cos }^2}\theta }}{{{{(2 + \cos \theta )}^2}}} = \dfrac{{\cos \theta (4 - \cos \theta )}}{{{{(2 + \cos \theta )}^2}}}$

In $\left[ {0,\dfrac{\pi }{2}} \right],\cos \theta  > 0$,

$4 > \cos \theta  \Rightarrow 4 - \cos \theta  > 0.$

$\therefore \cos \theta (4 - \cos \theta ) > 0$

${(2 + \cos \theta )^2} > 0$

$\Rightarrow \dfrac{{\cos \theta (4 - \cos \theta )}}{{{{(2 + \cos \theta )}^2}}} > 0$

$\Rightarrow \dfrac{{dy}}{{d\theta }} > 0$

So, $y$ is strictly increasing in $\left( {0,\dfrac{\pi }{2}} \right)$ The function is continuous at $x = 0$ and $x = \dfrac{\pi }{2}$.

So, $y$ is increasing in $\left[ {0,\dfrac{\pi }{2}} \right]$.

10. Prove that the logarithmic function is strictly increasing on $(0,\infty )$.

Ans: $f(x) = \log x$

$\therefore {f^\prime }(x) = \dfrac{1}{x}$

For $x > 0,{f^\prime }(x) = \dfrac{1}{x} > 0$

Thus, the logarithmic function is strictly increasing in interval $(0,\infty )$.

11.Prove that the function $f$ given by $f(x) = {x^2} - x + 1$ is neither strictly increasing nor strictly decreasing on $( - 1,1)$.

Ans: $f(x) = {x^2} - x + 1$

$\therefore {f^\prime }(x) = 2x - 1$

${f^\prime }(x) = 0 \Rightarrow x = \dfrac{1}{2}$

$x = \dfrac{1}{2}$ divides $( - 1,1)$ into $\left( { - 1,\dfrac{1}{2}} \right)$ and $\left( {\dfrac{1}{2},1} \right)$.

In $\left( { - 1,\dfrac{1}{2}} \right),$

${f^\prime }(x) = 2x - 1 < 0$

So, $f$ is strictly decreasing in $\left( { - 1,\dfrac{1}{2}} \right)$

$\operatorname{In} \left( {\dfrac{1}{2},1} \right),$

${f^\prime }(x) = 2x - 1 > 0$

So, $f$ is strictly increasing in interval $\left( {\dfrac{1}{2},1} \right)$. 

Thus, $f$ is neither strictly increasing nor strictly decreasing in interval $( - 1,1)$.

12. Which of the following functions are strictly decreasing on $\left( {0,\dfrac{\pi }{2}} \right)$ ?

1. $\cos x$

2. $\cos 2{\text{x}}$

3. $\cos 3x$

4. $\tan {\text{x}}$

Ans: 

1. ${f_1}(x) = \cos x$.

$\therefore f_1^\prime (x) =  - \sin x$

$\operatorname{In} \left( {0,\dfrac{\pi }{2}} \right),f_1^\prime (x) =  - \sin x < 0.$

$\therefore {f_1}(x) = \cos x$ is strictly decreasing in $\left( {0,\dfrac{\pi }{2}} \right)$.

2. ${f_2}(x) = \cos 2x$

$\therefore f_2^\prime (x) =  - 2\sin 2x$

$0 < x < \dfrac{\pi }{2}$

$\Rightarrow 0 < 2x < \pi $

$\Rightarrow \sin 2x > 0 \Rightarrow  - 2\sin 2x < 0$

$\therefore f_2^\prime (x) =  - 2\sin 2x < 0$ in $\left( {0,\dfrac{\pi }{2}} \right)$

$\therefore {f_2}(x) = \cos 2x$ is strictly decreasing in $\left( {0,\dfrac{\pi }{2}} \right)$.

3. ${f_3}(x) = \cos 3x$

$\therefore f_3^\prime (x) =  - 3\sin 3x$

$f_3^\prime (x) = 0$

$\Rightarrow \sin 3x = 0 \Rightarrow 3x = \pi $, as $x \in \left( {0,\dfrac{\pi }{2}} \right)$

$\Rightarrow x = \dfrac{\pi }{3}$

$x = \dfrac{\pi }{3}$ divides $\left( {0,\dfrac{\pi }{2}} \right)$ into $\left( {0,\dfrac{\pi }{3}} \right)$ and $\left( {\dfrac{\pi }{3},\dfrac{\pi }{2}} \right)$.

In $\left( {0,\dfrac{\pi }{3}} \right),{f_3}(x) =  - 3\sin 3x < 0\quad \left[ {0 < x < \dfrac{\pi }{3} \Rightarrow 0 < 3x < \pi } \right]$

$\therefore {f_3}$ is strictly decreasing in $\left( {0,\dfrac{\pi }{3}} \right)$.

$\operatorname{In} \left( {\dfrac{\pi }{3},\dfrac{\pi }{2}} \right),{f_3}(x) =  - 3\sin 3x > 0\quad \left[ {\dfrac{\pi }{3} < x < \dfrac{\pi }{2} \Rightarrow \pi  < 3x < \dfrac{{3\pi }}{2}} \right]$

$\therefore {f_3}$ is strictly increasing in $\left( {\dfrac{\pi }{3},\dfrac{\pi }{2}} \right)$.

So, ${f_3}$ is neither increasing nor decreasing in interval $\left( {0,\dfrac{\pi }{2}} \right)$.

4. ${f_4}(x) = \tan x$

$\therefore f_4^\prime (x) = {\sec ^2}x$

In $\left( {0,\dfrac{\pi }{2}} \right),f_4^\prime (x) = {\sec ^2}x > 0.$

$\therefore {f_4}$ is strictly increasing in $\left( {0,\dfrac{\pi }{2}} \right)$.

So, the correct answers are ${\mathbf{A}}$ and ${\mathbf{B}}$.

13. On which of the following intervals is the function $f$ is given by $f(x) = {x^{100}} + \sin x - 1$ strictly decreasing?

1. $(0,1)$

2. $\left( {\dfrac{\pi }{2},\pi } \right)$

3. $\left( {0,\dfrac{\pi }{2}} \right)$

4. None of these

Ans: $f(x) = {x^{100}} + \sin x - 1$

$\therefore {f^\prime }(x) = 100{x^{99}} + \cos x$

In $(0,1),\cos x > 0$ and $100{x^{99}} > 0$

$\therefore {f^\prime }(x) > 0$

So, $f$ is strictly increasing in $(0,1)$.

In $\left( {\dfrac{\pi }{2},\pi } \right),\cos x < 0$ and $100{x^{99}} > 0$

$100{x^{99}} > \cos x$

$\therefore {f^\prime }(x) > 0$ in $\left( {\dfrac{\pi }{2},\pi } \right)$

So, $f$ is strictly increasing in interval $\left( {\dfrac{\pi }{2},\pi } \right)$. In interval $\left( {0,\dfrac{\pi }{2}} \right),\cos x > 0$ and $100{x^{99}} > 0$. $\therefore 100{x^{99}} + \cos x > 0$

$\Rightarrow {f^\prime }(x) > 0$ on $\left( {0,\dfrac{\pi }{2}} \right)$

$\therefore f$ is strictly increasing in interval $\left( {0,\dfrac{\pi }{2}} \right)$. Hence, $f$ is strictly decreasing in none of the intervals. 

The correct answer is ${\mathbf{D}}$.

14. Find the least value of $a$ such that the function $f$ given $f(x) = {x^2} + ax + 1$ is strictly increasing on $(1,2)$.

Ans: $f(x) = {x^2} + ax + 1$

$\therefore {f^\prime }(x) = 2x + a$

${f^\prime }(x) > 0$ in $(1,2)$

$\Rightarrow 2x + a > 0$

$\Rightarrow 2x >  - a$

$\Rightarrow x > \dfrac{{ - a}}{2}$

So, we need to find the smallest value of a such that

$x > \dfrac{{ - a}}{2}$, when $x \in (1,2)$.

$\Rightarrow x > \dfrac{{ - a}}{2}($ when $1 < x < 2)$

$\dfrac{{ - a}}{2} = 1 \Rightarrow a =  - 2$

Hence, the required value of $a$ is $ - 2$.

15. Let ${\mathbf{I}}$ be any interval disjoint from $( - 1,1)$, prove that the function $f$ given by $f(x) = x + \dfrac{1}{x}$ is strictly increasing on I. 

Ans: $f(x) = x + \dfrac{1}{x}$

$\therefore {f^\prime }(x) = 1 - \dfrac{1}{{{x^2}}}$

${f^\prime }(x) = 0$

$\Rightarrow \dfrac{1}{{{x^2}}}$

$\Rightarrow x =  \pm 1$

$x = 1$ and $x =  - 1$ divide the real line in intervals $( - \infty ,1),( - 1,1)$ and $(1,\infty )$. In $( - 1,1)$, $ - 1 < x < 1$

$\Rightarrow {x^2} < 1$

$\Rightarrow 1 < \dfrac{1}{{{x^2}}},x \ne 0$

$\Rightarrow 1 - \dfrac{1}{{{x^2}}} < 0,x \ne 0$

$\therefore {f^\prime }(x) = 1 - \dfrac{1}{{{x^2}}} < 0$ on $( - 1,1)\~\{ 0\} $.

In $( - \infty , - 1)$ and $(1,\infty )$,

$x <  - 1$ or $1 < x$

$\Rightarrow {x^2} > 1$

$\Rightarrow 1 > \dfrac{1}{{{x^2}}}$

$\Rightarrow 1 - \dfrac{1}{{{x^2}}} > 0$

$\therefore {f^\prime }(x) = 1 - \dfrac{1}{{{x^2}}} > 0$ on $( - \infty , - 1)$ and $(1,\infty )$.

$\therefore f$ is strictly increasing on $( - \infty ,1)$ and $(1,\infty )$. 

Hence, $f$ is strictly increasing in ${\mathbf{I}} - ( - 1,1)$.

16. Prove that the function $f$ given by $f(x) = \log \sin x$ is strictly increasing on $\left( {0,\dfrac{\pi }{2}} \right)$ and strictly decreasing on $\left( {\dfrac{\pi }{2},\pi } \right)$

Ans: $f(x) = \log \sin x$

$\therefore {f^\prime }(x) = \dfrac{1}{{\sin x}}\cos x = \cot x$

$\operatorname{In} \left( {0,\dfrac{\pi }{2}} \right),{f^\prime }(x) = \cot x > 0$

$\therefore f$ is strictly increasing in $\left( {0,\dfrac{\pi }{2}} \right)$. $\operatorname{In} \left( {\dfrac{\pi }{2},\pi } \right),{f^\prime }(x) = \cot x < 0$

$\therefore f$ is strictly decreasing in $\left( {\dfrac{\pi }{2},\pi } \right)$.

17. Prove that the function $f$ given by $f(x) = \log \cos x$ is strictly decreasing on $\left( {0,\dfrac{\pi }{2}} \right)$ and strictly increasing on $\left( {\dfrac{\pi }{2},\pi } \right)$

Ans: $f(x) = \log \cos x$

$\therefore {f^\prime }(x) = \dfrac{1}{{\cos x}}( - \sin x) =  - \tan x$

$\operatorname{In} \left( {0,\dfrac{\pi }{2}} \right),\tan x > 0 \Rightarrow  - \tan x < 0$.

$\therefore {f^\prime }(x) =  < 0$ on $\left( {0,\dfrac{\pi }{2}} \right)$

$\therefore f$ is strictly decreasing on $\left( {0,\dfrac{\pi }{2}} \right)$.

In $\left( {\dfrac{\pi }{2},\pi } \right),\tan x < 0$

$\Rightarrow  - \tan x > 0$

$\therefore {f^\prime }(x) > 0$ on $\left( {\dfrac{\pi }{2},\pi } \right)$

$\therefore f$ is strictly increasing on $\left( {\dfrac{\pi }{2},\pi } \right)$.

18. Prove that the function given by $f(x) = {x^3} - 3{x^2} + 3x = 100$ is increasing in ${\mathbf{R}}$. 

Ans: $f(x) = {x^3} - 3{x^2} + 3x = 100$

${f^\prime }(x) = 3{x^2} - 6x + 3$

$= 3\left( {{x^2} - 2x + 1} \right)$

$= 3{(x - 1)^2}$

For $x \in {\mathbf{R}}{(x - 1)^2} \geqslant 0$.

So ${f^\prime }(x)$ is always positive in ${\mathbf{R}}$.

So, the $f$ is increasing in ${\mathbf{R}}$.

19. The interval in which $y = {x^2}{e^{ - x}}$ is increasing is

1. $( - \infty ,\infty )$

2. $( - 2,0)$

3. $(2,\infty )$

4. $(0,2)$ 

Ans: $y = {x^2}{e^{ - x}}$

$\therefore \dfrac{{dy}}{{dx}} = 2x{e^{ - x}} - {x^2}{e^{ - x}} = x{e^{ - x}}(2 - x)$

$\dfrac{{dy}}{{dx}} = 0$

$\Rightarrow x = 0$ and $x = 2$

In $( - \infty ,0)$ and $(2,\infty ),{f^\prime }(x) < 0$ as ${e^{ - x}}$ is always positive. 

$\therefore f$ is decreasing on $( - \infty ,0)$ and $(2,\infty )$. In $(0,2),{f^\prime }(x) > 0$.

$\therefore f$ is strictly increasing on $(0,2)$. So, $f$ is strictly increasing in $(0,2)$. 

The correct answer is ${\mathbf{D}}$.

Exercise 6.3

1. Find the slope of the tangent to the curve $y = 3{x^4} - 4x$ at $x = 4$.

Ans: $y = 3{x^4} - 4x$

Differentiate

${\left. {{{\left. {\dfrac{{dy}}{{dx}}} \right]}_{x = 4}} = \dfrac{d}{{dx}}\left( {3{x^4} - 4x} \right) = 12{x^3} - 4} \right]_{x = 4}}$

$= 12{(4)^3} - 4 = 12(64) - 4$

$= 764$

2. Find the slope of the tangent to the curve $y = \dfrac{{x - 1}}{{x - 2}},x \ne 2$ at $x = 10$.

Ans: $y = \dfrac{{x - 1}}{{x - 2}},$

$\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {\dfrac{{x - 1}}{{x - 2}}} \right) = \dfrac{{(x - 2)(1) - (x - 1)(1)}}{{{{(x - 2)}^2}}}$

$= \dfrac{{x - 2 - x + 1}}{{{{(x - 2)}^2}}} = \dfrac{{ - 1}}{{{{(x - 2)}^2}}}$

${\left. {{{\left. {\therefore \dfrac{{dy}}{{dx}}} \right]}_{x = 10}} = \dfrac{{ - 1}}{{{{(x - 2)}^2}}}} \right]_{x = 10}}$

$= \dfrac{{ - 1}}{{{{(10 - 2)}^2}}} = \dfrac{{ - 1}}{{64}}$

3. Find the slope of the tangent to curve $y = {x^3} - x + 1$ at the point whose ${\text{X}}$-coordinate is \[2.\]

Ans: $y = {x^3} - x + 1$

$\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{x^3} - x + 1} \right) = 3{x^2} - 1$

${\left. {{{\left. {\dfrac{{dy}}{{dx}}} \right]}_{x = 2}} = 3{x^2} - 1} \right]_{x = 2}}$

$= 3{(2)^2} - 1$

${\left. {{{\left. {\dfrac{{dy}}{{dx}}} \right]}_{x = 2}} = 3{x^2} - 1} \right]_{x = 2}}$

$= 3{(2)^2} - 1$

$= 12 - 1 = 11$

4. Find the slope of the tangent to the curve $y = {x^3} - 3x + 2$ at the point whose $X$-coordinate is 3 .

Ans: $y = {x^3} - 3x + 2$

Differentiate

$\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{x^3} - 3x + 2} \right) = 3{x^2} - 3$

${\left. {{{\left. {\therefore \dfrac{{dy}}{{dx}}} \right]}_{x = 3}} = 3{x^2} - 3} \right]_{x = 3}}$

$= 3{(3)^2} - 3 = 27 - 3 = 24$

5. Find the slope of the normal to the curve $x = a{\cos ^3}\theta ,y = a{\sin ^3}\theta $ at $\theta  = \dfrac{\pi }{4}$.

Ans: $x = a{\cos ^3}\theta ,y = a{\sin ^3}\theta $

$\dfrac{{dx}}{{d\theta }} = \dfrac{d}{{d\theta }}\left( {a{{\cos }^3}\theta } \right) =  - 3a{\cos ^2}\theta \sin \theta $

$\dfrac{{dy}}{{d\theta }} = \dfrac{d}{{d\theta }}\left( {a{{\sin }^3}\theta } \right) = 3a{\sin ^2}\theta (\cos \theta )$

$\therefore \dfrac{{dy}}{{dx}} = \dfrac{{\left( {\dfrac{{dy}}{{d\theta }}} \right)}}{{\left( {\dfrac{{dx}}{{d\theta }}} \right)}} =  = \dfrac{{3a{{\sin }^2}\theta \cos \theta }}{{ - 3a{{\cos }^2}\theta \sin \theta }}$

$=  - \dfrac{{\sin \theta }}{{\cos \theta }} =  - \tan \theta $

${\left. {{{\left. {\therefore \dfrac{{dy}}{{dx}}} \right]}_{\theta  = \dfrac{\pi }{4}}} =  - \tan \theta } \right]_{\theta  = \dfrac{\pi }{4}}}$

$=  - \tan \dfrac{\pi }{4} =  - 1$

Slope of normal at $\theta  = \dfrac{\pi }{4} = \dfrac{{ - 1}}{{ - 1}} = 1$

6. Find the slope of the normal to the curve $x = 1 - a\sin \theta $ and $y = b{\cos ^2}\theta $ at $\theta  = \dfrac{\pi }{2}$.

Ans: curve $x = 1 - a\sin \theta $ and $y = b{\cos ^2}\theta $ 

$\dfrac{{dx}}{{d\theta }} = \dfrac{d}{{d\theta }}(1 - a\sin \theta ) =  - a\cos \theta $

$\dfrac{{dy}}{{d\theta }} = \dfrac{d}{{d\theta }}\left( {b{{\cos }^2}\theta } \right) =  - 2b\sin \theta \cos \theta $

$\therefore \dfrac{{dy}}{{dx}} = \dfrac{{\left( {\dfrac{{dy}}{{d\theta }}} \right)}}{{\left( {\dfrac{{dx}}{{d\theta }}} \right)}} = \dfrac{{ - 2b\sin \theta \cos \theta }}{{ - a\cos \theta }} = \dfrac{{2b}}{a}\sin \theta $

${\left. {{{\left. {\dfrac{{dy}}{{dx}}} \right]}_{\theta  = \dfrac{\pi }{2}}} = \dfrac{{2b}}{a}\sin \theta } \right]_{\theta  - \dfrac{\pi }{2}}}$

$= \dfrac{{2b}}{a}\sin \dfrac{\pi }{2} = \dfrac{{2b}}{a}$

Slope of normal at $\theta  = \dfrac{\pi }{2}$

$= \dfrac{{ - 1}}{{\left( {\dfrac{{2b}}{a}} \right)}} =  - \dfrac{a}{{2b}}$

7. Find the points at which tangent to the curve $y = {x^3} - 3{x^2} - 9x + 7$ is parallel to the $X$ - axis. 

Ans: $y = {x^3} - 3{x^2} - 9x + 7$

$\therefore \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{x^3} - 3{x^2} - 9x + 7} \right) = 3{x^2} - 6x - 9$

Since tangent is parallel to the ${\text{X}} - $ axis, slope $= 0$. 

$\therefore 3{x^2} - 6x - 9 = 0$

$\Rightarrow {x^2} - 2x - 3 = 0$

$\Rightarrow (x - 3)(x + 1) = 0$

$\Rightarrow x = 3$ or $x =  - 1$

When $x = 3,$

$y = {(3)^3} - 9(3) + 7$

$= 27 - 27 - 27 + 7 =  - 20$

When $x = 1,$

$y = {( - 1)^3} - 3{( - 1)^2} - 9( - 1) + 7$

$=  - 1 - 3 + 9 + 7 = 12$

Hence, the required points are $(3, - 20)$ and $( - 1,12)$.

8. Find a point on the curve $y = {(x - 2)^2}$ at which the tangent is parallel to the chord joining the points $(2,0)$ and $(4,4)$.

Ans: Slope of chord $= \dfrac{{4 - 0}}{{4 - 2}} = \dfrac{4}{2} = 2$

Slope of tangent $= \dfrac{{dy}}{{dx}} = 2(x - 2)$

$\therefore 2(x - 2) = 2$

$\Rightarrow x - 2 = 1 \Rightarrow x = 3$

When $x = 3,$

$y = {(3 - 2)^2} = 1$

Hence, the point is $(3,1)$.

9. Find the point on the curve $y = {x^3} - 11x + 5$ at which the tangent is $y = x - 11$.

Ans: Equation of tangent is $y = x - 11$

$\therefore $ Slope of the tangent $= 1$ 

$\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{x^3} - 11x + 5} \right) = 3{x^2} - 11$

$\therefore 3{x^2} - 11 = 1$

$\Rightarrow 3{x^2} = 12$

$\Rightarrow {x^2} = 4$

$\Rightarrow x =  \pm 2$

When $x = 2,$

$y = {(2)^3} - 11(2) + 5 = 8 - 22 + 5 =  - 9$

When $x =  - 2,$

$y = {( - 2)^3} - 11( - 2) + 5 =  - 8 + 22 + 5 = 19$

So, the points are $(2, - 9)$ and $( - 2,19)$.

10. Find the equation of all lines having slope $ - 1$ that are tangents to the curve $y = \dfrac{1}{{x - 1}},x \ne 1$ 

Ans: $y = \dfrac{1}{{x - 1}}$

$\dfrac{{dy}}{{dx}} = \dfrac{{ - 1}}{{{{(x - 1)}^2}}}$

$\dfrac{{ - 1}}{{{{(x - 1)}^2}}} =  - 1$

$\Rightarrow {(x - 1)^2} = 1$

$\Rightarrow x - 1 =  \pm 1$

$\Rightarrow x = 2,0$

$x = 0,y =  - 1$ and $x = 2,y = 1$

$y - ( - 1) =  - 1(x - 0)$

$\Rightarrow y + 1 =  - x$

$\Rightarrow y + x + 1 = 0$

$\Rightarrow y - 1 =  - 1(x - 2)$

$\Rightarrow y - 1 = x + 2$

$\Rightarrow y + x - 3 = 0$

So, the equations of the required lines are $y + x + 1 = 0$ and $y + x - 3 = 0$.

11. Find the equation of all lines having slope 2 which are tangents to the curve $y = \dfrac{1}{{x - 3}},x \ne 3.$ 

Ans: $y = \dfrac{1}{{x - 3}}$

$\dfrac{{dy}}{{dx}} = \dfrac{{ - 1}}{{{{(x - 3)}^2}}}$

$\dfrac{{ - 1}}{{{{(x - 3)}^2}}} = 2$

$\Rightarrow 2{(x - 3)^2} =  - 1$

$\Rightarrow {(x - 3)^2} = \dfrac{{ - 1}}{2}$

Which is not possible.

So, there is no tangent to the curve of slope \[2.\]

12. Find the equations of all lines having slope 0 which are tangent to the curve $y = \dfrac{1}{{{x^2} - 2x + 3}}$. 

Ans: $y = \dfrac{1}{{{x^2} - 2x + 3}}$

$\dfrac{{dy}}{{dx}} = \dfrac{{ - (2x - 2)}}{{{{\left( {{x^2} - 2x + 3} \right)}^2}}} = \dfrac{{ - 2(x - 1)}}{{{{\left( {{x^2} - 2x + 3} \right)}^2}}}$

$\dfrac{{ - 2(x - 1)}}{{{{\left( {{x^2} - 2x + 3} \right)}^2}}} = 0$

$\Rightarrow  - 2(x - 1) = 0$

$\Rightarrow x = 1$

When $x = 1,$

$y = \dfrac{1}{{1 - 2 + 3}} = \dfrac{1}{2}$

$y - \dfrac{1}{2} = 0(x - 1)$

$\Rightarrow y - \dfrac{1}{2} = 0$

$\Rightarrow y = \dfrac{1}{2}$

So, the equation of the line is $y = \dfrac{1}{2}$.

13. Find points on the curve $\dfrac{{{x^2}}}{9} + \dfrac{{{y^2}}}{{16}} = 1$ at which the tangents are

i. Parallel to $x$-axis 

ii. Parallel to $y$-axis

Ans: $\dfrac{{{x^2}}}{9} + \dfrac{{{y^2}}}{{16}} = 1$

Differentiate 

$\dfrac{{2x}}{9} + \dfrac{{2y}}{{16}}\dfrac{{dy}}{{dx}} = 0$

$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{ - 16x}}{{9y}}$

(i) $\dfrac{{dy}}{{dx}} = \dfrac{{ - 16x}}{{9y}} = 0$

$\Rightarrow x = 0$

$\dfrac{{{x^2}}}{9} + \dfrac{{{y^2}}}{{16}} = 1$ for $x = 0$

${y^2} = 16 \Rightarrow y =  \pm 4$

So, the points are $(0,4)$ and $(0, - 4)$.

(ii) $\dfrac{{dx}}{{dy}} = 0$

$\Rightarrow \dfrac{{ - 1}}{{\left( {\dfrac{{ - 16x}}{{9y}}} \right)}}$

$= \dfrac{{9y}}{{16x}} = 0$

$\Rightarrow y = 0$

$\dfrac{{{x^2}}}{9} + \dfrac{{{y^2}}}{{16}} = 1$ for $y = 0.$

$\Rightarrow x =  \pm 3$

So, the points are $(3,0)$ and $( - 3,0)$.

14. Find the equations of the tangents and normal to the given curves at the indicated points:

1. $\quad y = {x^4} - 6{x^3} + 13{x^2} - 10x + 5$ at $(0,5)$

2. $\quad y = {x^4} - 6{x^3} + 13{x^2} - 10x + 5$ at $(1,3)$

3. $\quad y = {x^3}\quad $ at $(1,1)$

4. $\quad y = {x^2}$ at $(0,0)$

5. $\quad x = \cos t,y = \sin t$ at $t = \dfrac{\pi }{4}$

Ans: 

1. $\dfrac{{dy}}{{dx}} = 4{x^3} - 18{x^2} + 26x - 10$

${\left. {\dfrac{{dy}}{{dx}}} \right]_{(0,5)}} =  - 10$

Slope of tangent at $(0,5)$ is $ - 10$.

$y - 5 =  - 10(x - 0)$

$\Rightarrow y - 5 =  - 10x$

$\Rightarrow 10x + y = 5$

Slope of normal at $(0,5)$ is $\dfrac{{ - 1}}{{ - 10}} = \dfrac{1}{{10}}$ $y - 5 = \dfrac{1}{{10}}(x - 0)$

$\Rightarrow 10y - 50 = x$

$\Rightarrow x - 10y + 50 = 0$

2. $\dfrac{{dy}}{{dx}} = 4{x^3} - 18{x^2} + 26x - 10$

${\left. {\dfrac{{dy}}{{dx}}} \right]_{(1,3)}} = 4 - 18 + 26 - 10 = 2$

Slope of tangent at $(1,3)$ is \[2.\]

$y - 3 = 2(x - 1)$

$\Rightarrow y - 3 = 2x - 2$

$\Rightarrow y = 2x + 1$

Slope of normal at $(1,3)$ is $ - \dfrac{1}{2}$

$y - 3 = \dfrac{1}{2}(x - 1)$

$\Rightarrow 2y - 6 = x + 1$

$\Rightarrow x + 2y - 7 = 0$

3. $\dfrac{{dy}}{{dx}} = 3{x^2}$

${\left. {\dfrac{{dy}}{{dx}}} \right]_{(1,1)}} = 3{(1)^2} = 3$

Slope of tangent at $(1,1)$ is \[3.\]

 $y - 1 = 3(x - 1)$

$\Rightarrow y = 3x - 2$

Slope of normal at $(1,1)$ is $ - \dfrac{1}{3}$.

 $y - 1 = \dfrac{{ - 1}}{3}(x - 1)$

$\Rightarrow 3y - 3 =  - x + 1$

$\Rightarrow x + 3y - 4 = 0$

4. $\dfrac{{dy}}{{dx}} = 2x$

${\left. {\dfrac{{dy}}{{dx}}} \right]_{(0,0)}} = 0$

Slope of tangent at $(0,0)$ is \[0.\]

$y - 0 = 0(x - 0)$

$\Rightarrow y = 0$

Slope of normal at $(0,0)$ is $ - \dfrac{1}{0}$, which is undefined.

$\therefore x = 0$

5. $x = \cos t$

$y = \sin t$

$\therefore \dfrac{{dx}}{{dt}} =  - \sin t$

$\dfrac{{dy}}{{dt}} = \cos t$

$\dfrac{{dy}}{{dx}} = \dfrac{{\left( {\dfrac{{dy}}{{dt}}} \right)}}{{\left( {\dfrac{{dx}}{{dt}}} \right)}} = \dfrac{{\cos t}}{{ - \sin t}} =  - \cot t$

${\left. {\dfrac{{dy}}{{dx}}} \right]_{t = \dfrac{\pi }{4}}} =  - \cot t =  - 1$

Slope of tangent at $t = \dfrac{\pi }{4}$ is $ - 1$. 

$t = \dfrac{\pi }{4},{\kern 1pt} {\kern 1pt} {\kern 1pt} \,{\kern 1pt} x = \dfrac{1}{{\sqrt 2 }},{\kern 1pt} \,{\kern 1pt} y = \dfrac{1}{{\sqrt 2 }}$

$y - \dfrac{1}{{\sqrt 2 }} =  - 1\left( {x - \dfrac{1}{{\sqrt 2 }}} \right)$

$\Rightarrow x + y - \dfrac{1}{{\sqrt 2 }} - \dfrac{1}{{\sqrt 2 }} = 0$

$\Rightarrow x + y - \sqrt 2  = 0$

Slope of normal at $t = \dfrac{\pi }{4}$ is $\dfrac{{ - 1}}{{ - 1}} = 1$ 

$y - \dfrac{1}{{\sqrt 2 }} = 1\left( {x - \dfrac{1}{{\sqrt 2 }}} \right)$

$\Rightarrow x = y$

15. Find the equation of the tangent line to the curve $y = {x^2} - 2x + 7$ which is

1. parallel to the line $2x - y + 9 = 0$

2. Perpendicular to the line $5y - 15x = 13$.

Ans:  $y = {x^2} - 2x + 7$

$\dfrac{{dy}}{{dx}} = 2x - 2.$

1. $2x - y + 9 = 0 \Rightarrow y = 2x + 9$

Slope of line $= 2$

$\therefore 2 = 2x - 2$

$\Rightarrow 2x = 4 \Rightarrow x = 2$

$x = 2$

$\Rightarrow y = 4 - 4 + 7 = 7$

Equation of tangent through $(2,7)$ is

$y - 7 = 2(x - 2)$

$\Rightarrow y - 2x - 3 = 0$

2. $5y - 15x = 13$

$\Rightarrow y = 3x + \dfrac{{13}}{5}$

Slope of line $= 3$

 $\therefore 2x - 2 = \dfrac{{ - 1}}{3}$

$\Rightarrow 2x = \dfrac{{ - 1}}{3} + 2$

$\Rightarrow 2x = \dfrac{5}{3}$

$\Rightarrow x = \dfrac{5}{6}$

$\Rightarrow y = \dfrac{{25}}{{36}} + \dfrac{{10}}{6} + 7$

$= \dfrac{{25 - 60 + 252}}{{36}}$

$= \dfrac{{217}}{{36}}$

Equation of tangent through $\left( {\dfrac{5}{6},\dfrac{{217}}{{36}}} \right)$ is 

$y - \dfrac{{217}}{{36}} = \dfrac{1}{3}\left( {x - \dfrac{5}{6}} \right)$

$\Rightarrow \dfrac{{36y - 217}}{{36}} = \dfrac{{ - 1}}{{18}}(6x - 5)$

$\Rightarrow 36y - 217 =  - 2(6x - 5)$

$\Rightarrow 36y - 217 =  - 12x + 10$

$\Rightarrow 36y + 12x - 227 = 0$

16. Show that the tangents to the curve $y = 7{x^3} + 11$ at the points where $x = 2$ and $x =  - 2$ are parallel. 

Ans: $y = 7{x^3} + 11$

$\therefore \dfrac{{dy}}{{dx}} = 21{x^2}$

${\left. {\dfrac{{dy}}{{dx}}} \right]_{x = 2}} = 21{(2)^2} = 84$

${\left. {\dfrac{{dy}}{{dx}}} \right]_{x =  - 2}} = 21{( - 2)^2} = 84$

Clearly, the tangents are parallel.

17. Find the points on the curve $y = {x^3}$ at which the slope of the tangent is equal to the $y$-coordinate of the point. 

Ans: $y = {x^3}$

$\therefore \dfrac{{dy}}{{dx}} = 3{x^2}$

According to the question, $y = \dfrac{{dy}}{{dx}} = 3{x^2}$ 

Also, $y = {x^3}$ 

$\therefore 3{x^2} = {x^3}$

${x^2}(x - 3) = 0$

$x = 0,x = 3$

$x = 0,$ $y = 0$and $x = 3,$$y = 3{(3)^2} = 27$

So, the points are $(0,0)$ and $(3,27)$.

18. For the curve $y = 4{x^3} - 2{x^5}$, find all the points at which the tangents passes through the origin.

Ans: $y = 4{x^3} - 2{x^5}$

$\dfrac{{dy}}{{dx}} = 12{x^2} - 10{x^4}$

Equation of tangent through $({\text{X}},{\text{Y}})$ is

$Y - y = \left( {12{x^2} - 10{x^4}} \right)(X - x)$

For passing through origin, ${\text{X}} = 0,$${\text{Y}} = 0$

$ - y = \left( {12{x^2} - 10{x^4}} \right)( - x)$

$y = 12{x^3} - 10{x^5}$

Also, $y = 4{x^3} - 2{x^5}$

$\therefore 12{x^3} - 10{x^5} = 4{x^3} - 2{x^5}$

$\Rightarrow 8{x^5} - 8{x^3} = 0$

$\Rightarrow {x^5} - {x^3} = 0$

$\Rightarrow {x^3}\left( {{x^2} - 1} \right) = 0$

$\Rightarrow x = 0, \pm 1$

When $x = 0,$

$y = 4{(0)^3} - 2{(0)^5} = 0.$

When $x = 1,$

$y = 4{(1)^3} - 2{(1)^5} = 2$

When $x =  - 1,$

$y = 4{( - 1)^3} - 2{( - 1)^5} =  - 2$

So, the points are $(0,0),(1,2)$ and $( - 1, - 2)$.

19. Find the points on the curve ${x^2} + {y^2} - 2x - 3 = 0$  at which the tangents are parallel to the $x$-axis. 

Ans: ${x^2} + {y^2} - 2x - 3 = 0$

$2x + 2y\dfrac{{dy}}{{dx}} - 2 = 0$

$\Rightarrow y\dfrac{{dy}}{{dx}} = 1 - x$

$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{1 - x}}{y}$

For parallel to $X$ axis,

 $\therefore \dfrac{{1 - x}}{y} = 0$

$\Rightarrow 1 - x = 0 \Rightarrow x = 1$

${x^2} + {y^2} - 2x - 3 = 0$

$\Rightarrow {y^2} = 4,y =  \pm 2$

So, the points are $(1,2)$ and $(1, - 2)$.

20. Find the equation of the normal at the point $\left( {a{m^2},a{m^3}} \right)$ for the curve $a{y^2} = {x^3}$.

Ans: $a{y^2} = {x^3}$

$2ay\dfrac{{dy}}{{dx}} = 3{x^2}$

$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{3{x^2}}}{{2ay}}$

Slope of tangent at $\left( {{\text{a}}{{\text{m}}^2},{\text{a}}{{\text{m}}^3}} \right)$ is

${\left. {\dfrac{{dy}}{{dx}}} \right]_{\left( {a{m^2},a{m^3}} \right)}} = \dfrac{{3{{\left( {a{m^2}} \right)}^2}}}{{2a\left( {a{m^3}} \right)}} = \dfrac{{3{a^2}{m^4}}}{{2{a^2}{m^3}}} = \dfrac{{3m}}{2}$

Slope of normal $= \dfrac{{ - 2}}{{3m}}$

$y - a{m^3} = \dfrac{{ - 2}}{{3m}}\left( {x - a{m^2}} \right)$

$\Rightarrow 3my - 3a{m^4} =  - 2x + 2a{m^2}$

$\Rightarrow 2x + 3my - a{m^2}\left( {2 + 3{m^2}} \right) = 0$

21. Find the equation of the normal to the curve $y = {x^3} + 2x + 6$ which are parallel to the line $x + 14y + 4 = 0.$

Ans: $y = {x^3} + 2x + 6$

$\dfrac{{dy}}{{dx}} = 3{x^2} + 2$

Slope of the normal $= \dfrac{{ - 1}}{{3{x^2} + 2}}$

$x + 14y + 4 = 0$

$\Rightarrow y =  - \dfrac{1}{{14}}x - \dfrac{4}{{14}}$

$\therefore \dfrac{{ - 1}}{{3{x^2} + 2}} = \dfrac{{ - 1}}{{14}}$

$\Rightarrow 3{x^2} + 2 = 14$

$\Rightarrow 3{x^2} = 12$

$\Rightarrow {x^2} = 4$

$\Rightarrow x =  \pm 2$

When $x = 2,$

$y = 8 + 4 + 6 = 18$

When $x =  - 2,$

$y =  - 8 - 4 + 6 =  - 6$

Equation of normal through $(2,18)$ is

$y - 18 = \dfrac{{ - 1}}{{14}}(x - 2)$

$\Rightarrow 14y - 252 = x + 2$

$\Rightarrow x + 14y - 254 = 0$

Equation of normal through $( - 2, - 6)$ is

$y - ( - 6) = \dfrac{{ - 1}}{{14}}[x - ( - 2)]$

$\Rightarrow y + 6 = \dfrac{{ - 1}}{{14}}(x + 2)$

$\Rightarrow 14y + 84 =  - x - 2$

$\Rightarrow x + 14y + 86 = 0$

22. Find the equations of the tangent and normal to the parabola ${y^2} = 4ax$ at the point $\left( {a{t^2},2at} \right)$.

Ans: ${y^2} = 4ax$

$2y\dfrac{{dy}}{{dx}} = 4a$

$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{2a}}{y}$

${\left. {\dfrac{{dy}}{{dx}}} \right]_{\left( {a{t^2},2at} \right)}} = \dfrac{{2a}}{{2at}} = \dfrac{1}{t}$

Slope of tangent $= \dfrac{1}{t}$ Equation of tangent is $y - 2at = \dfrac{1}{t}\left( {x - a{t^2}} \right)$

$\Rightarrow ty - 2a{t^2} = x - a{t^2}$

$\Rightarrow ty = x + a{t^2}$

Slope of normal $=  - \dfrac{1}{{\left( {\dfrac{1}{t}} \right)}} =  - t$

Equation of normal is $y - 2at =  - t\left( {x - a{t^2}} \right)$

$\Rightarrow y - 2at =  - tx + a{t^3}$

$\Rightarrow y =  - tx + 2at + a{t^3}$

23. Prove that the curves $x = {y^2}$ and $xy = k$ cut at right angles if $8{k^2} = 1$ 

(Hint : Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.)

Ans: The curves are $x = {y^2}$ and $xy = k$

Putting $x = {y^2}$ in $xy = k$, ${y^3} = k \Rightarrow y = {k^{\dfrac{1}{3}}}$

$\therefore x = {k^{\dfrac{2}{3}}}$

So, the point of intersection is $\left( {{k^{\dfrac{2}{3}}},{k^{\dfrac{1}{3}}}} \right)$

Differentiating $x = {y^2}$,

$1 = 2y\dfrac{{dy}}{{dx}} \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{2y}}$

So, slope of tangent to $x = {y^2}$ at $\left( {{k^{\dfrac{2}{3}}},{k^{\dfrac{1}{3}}}} \right)$ is $\left. {\dfrac{{dy}}{{dx}}} \right]\left( {\dfrac{2}{{\left. {{k^{\dfrac{2}{3}}},{k^{\dfrac{1}{3}}}} \right)}} = \dfrac{1}{{2{k^{\dfrac{1}{3}}}}}} \right.$

Differentiating $xy = k$, 

$x\dfrac{{dy}}{{dx}} + y = 0 \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{ - y}}{x}$

Slope of tangent to $xy = k$ at $\left( {{k^{\dfrac{2}{3}}},{k^{\dfrac{1}{3}}}} \right)$ is 

$\left. {\dfrac{{dy}}{{dx}}} \right]\left( {\dfrac{2}{{\left. {{k^{\dfrac{2}{3}}},{k^{\dfrac{1}{3}}}} \right)}} = \dfrac{{ - y}}{x}} \right]\left( {\dfrac{2}{{\left. {{k^{\dfrac{2}{3}}},{k^{\dfrac{1}{3}}}} \right)}}} \right.$

$=  - \dfrac{{{k^{\dfrac{1}{3}}}}}{{{k^{\dfrac{2}{3}}}}} = \dfrac{{ - 1}}{{{k^{\dfrac{1}{3}}}}}$

$\left( {\dfrac{1}{{2{k^{\dfrac{1}{3}}}}}} \right)\left( {\dfrac{{ - 1}}{{{k^{\dfrac{1}{3}}}}}} \right) =  - 1$ for perpendicularity condition.

$\Rightarrow 2{k^{\dfrac{2}{3}}} = 1$

$\Rightarrow {\left( {2{k^{\dfrac{2}{3}}}} \right)^3} = {(1)^3}$

$\Rightarrow 8{k^2} = 1$

So, the curves intersect at right angles if $8{k^2} = 1$.

24. Find the equations of the tangent and normal to the hyperbola $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$ at the point $\left( {{x_0}{y_0}} \right)$ 

Ans: $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$

 $\dfrac{{2x}}{{{a^2}}} - \dfrac{{2y}}{{{b^2}}}\dfrac{{dy}}{{dx}} = 0$

$\Rightarrow \dfrac{{2y}}{{{b^2}}}\dfrac{{dy}}{{dx}} = \dfrac{{2x}}{{{a^2}}}$

$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{{b^2}x}}{{{a^2}y}}$

Slope of tangent at $\left( {{x_0},{y_0}} \right)$ is ${\left. {\dfrac{{dy}}{{dx}}} \right]_{\left( {{x_0},{y_0}} \right)}} = \dfrac{{{b^2}{x_0}}}{{{a^2}{y_0}}}$

Equation of tangent at $\left( {{x_0},{y_0}} \right)$ is $y - {y_0} = \dfrac{{{b^2}{x_0}}}{{{a^2}{y_0}}}\left( {x - {x_0}} \right)$

$\Rightarrow {a^2}y{y_0} - {a^2}y_0^2 = {b^2}x{x_0} - {b^2}x_0^2$

$\Rightarrow {b^2}x{x_0} - {a^2}y{y_0} - {b^2}x_0^2 + {a^2}y_0^2 = 0$

$\Rightarrow \dfrac{{x{x_0}}}{{{a^2}}} - \dfrac{{y{y_0}}}{{{b^2}}} - \left( {\dfrac{{x_0^2}}{{{a^2}}} - \dfrac{{y_0^2}}{{{b^2}}}} \right) = 0$

$\Rightarrow \dfrac{{x{x_0}}}{{{a^2}}} - \dfrac{{y{y_0}}}{{{b^2}}} - 1 = 0$

$\Rightarrow \dfrac{{x{x_0}}}{{{a^2}}} - \dfrac{{y{y_0}}}{{{b^2}}} = 1$

Slope of normal at $\left( {{x_0},{y_0}} \right) = \dfrac{{ - {a^2}{y_0}}}{{{b^2}{x_0}}}$

Equation of normal at $\left( {{x_0},{y_0}} \right)$ is

$y - {y_0} = \dfrac{{ - {a^2}{y_0}}}{{{b^2}{x_0}}}\left( {x - {x_0}} \right)$

$\Rightarrow \dfrac{{y - {y_0}}}{{{a^2}{y_0}}} = \dfrac{{ - \left( {x - {x_0}} \right)}}{{{b^2}{x_0}}}$

$\Rightarrow \dfrac{{y - {y_0}}}{{{a^2}{y_0}}} - \dfrac{{x - {x_0}}}{{{b^2}{x_0}}} = 0$

25. Find the equation of the tangent to the curve $y = \sqrt {3x - 2} $ which is parallel to the line $4x - 2y + 5 = 0$

Ans: Slope of tangent at $(x,y)$ is $\dfrac{{dy}}{{dx}} = \dfrac{3}{{2\sqrt {3x - 2} }}$

The given line is $4x - 2y + 5 = 0$.

$4x - 2y + 5 = 0 \Rightarrow y = 2x + \dfrac{5}{2}$

Slope of line $= 2$

$\dfrac{3}{{2\sqrt {3x - 2} }} = 2$

$\Rightarrow \sqrt {3x - 2}  = \dfrac{3}{4}$

$\Rightarrow 3x - 2 = \dfrac{9}{{16}}$

$\Rightarrow 3x = \dfrac{9}{{16}} + 2 = \dfrac{{41}}{{16}}$

$\Rightarrow x = \dfrac{{41}}{{48}}$

When $x = \dfrac{{41}}{{48}},$

$y = \sqrt {3\left( {\dfrac{{41}}{{48}}} \right) - 2}  = \sqrt {\dfrac{{41}}{{16}} - 2} $

$= \sqrt {\dfrac{{41 - 32}}{{16}}}  = \sqrt {\dfrac{9}{{16}}}  = \dfrac{3}{4}$

Equation of tangent through $\left( {\dfrac{{41}}{{48}},\dfrac{3}{4}} \right)$ is $y - \dfrac{3}{4} = 2\left( {x - \dfrac{{41}}{{48}}} \right)$

$\Rightarrow \dfrac{{4y - 3}}{4} = 2\left( {\dfrac{{48x - 41}}{{48}}} \right)$

$\Rightarrow 4y - 3 = \left( {\dfrac{{48x - 41}}{6}} \right)$

$\Rightarrow 24y - 18 = 48x - 41$

$\Rightarrow 48x - 24y = 23$

26. The slope of the normal to the curve $y = 2{x^2} + 3\sin x$ at $x = 0$ is

1. 3

2. $\dfrac{1}{3}$,

3. $ - 3$,

4. $ - \dfrac{1}{3}$

Ans: $y = 2{x^2} + 3\sin x$

${\left. {{{\left. {\dfrac{{dy}}{{dx}}} \right]}_{x = 0}} = 4x + 3\cos x} \right]_{x = 0}} = 0 + 3\cos 0 = 3$

Slope of normal $= \dfrac{{ - 1}}{3}$ 

The correct answer is ${\mathbf{D}}$.

27. The line $y = x + 1$ is a tangent to the curve ${y^2} = 4x$ at the point

1. $(1,2)$,

2. $(2,1),($ C) 

3. $(1, - 2)$,

4. $( - 1,2)$

Ans: ${y^2} = 4x$ 

 $2y\dfrac{{dy}}{{dx}} = 4 \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{2}{y}$

Given line is $y = x + 1$. Slope of line $= 1$.

$\dfrac{2}{y} = 1$

$\Rightarrow y = 2$

$y = x + 1$

$\Rightarrow x = y - 1$

$\Rightarrow x = 2 - 1 = 1$

So, line $y = x + 1$ is tangent to curve at point $(1,2)$.

The correct answer is ${\mathbf{A}}$.

Exercise 6.4

1. Using differentials, find the approximate value of each of the following up to 3 places of decimal.

(i) $\sqrt {25.3} $,

(ii) $\sqrt {49.5} $, (iii) $\sqrt {0.6} $,

(iv) ${(0.009)^{\dfrac{1}{3}}}$, (v) ${(0.999)^{\dfrac{1}{{10}}}}$,

(vi) ${(15)^{\dfrac{1}{4}}}$, (vii) ${(26)^{\dfrac{1}{3}}}$

(viii) ${(255)^{\dfrac{1}{4}}}($ ix $){(82)^{\dfrac{1}{4}}}({\mathbf{x}}){(401)^{\dfrac{1}{2}}}({\mathbf{xi}}){(0.0037)^{\dfrac{1}{2}}},\left( {} \right.$ xii) ${(26.57)^{\dfrac{1}{3}}}$,

(xiii) ${(81.5)^{\dfrac{1}{4}}}$, (xiv) ${(3.968)^{\dfrac{3}{2}}}$

(xv) ${(32.15)^{\dfrac{1}{5}}}$

Ans: (i) $\sqrt {25.3} $

$y = \sqrt x $. Let $x = 25$ and $\Delta x = 0.3$.

$\Delta y = \sqrt {x + \Delta x}  - \sqrt x $

$= \sqrt {25.3}  - \sqrt {25}  = \sqrt {25.3}  - 5$

$\Rightarrow \sqrt {25.3}  = \Delta y + 5$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{2\sqrt x }}(0.3)$

$= \dfrac{1}{{2\sqrt {25} }}(0.3) = 0.03$

$\sqrt {25.3}  \approx 0.03 + 5 = 5.03$

(ii) $\sqrt {49.5} $

$y = \sqrt x $. 

Let $x = 49$ and $\Delta x = 0.5$.

$\Delta y = \sqrt {x + \Delta x}  - \sqrt x $

$= \sqrt {49.5}  - \sqrt {49}  = \sqrt {49.5}  - 7$

$\Rightarrow \sqrt {49.5}  = 7 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{2\sqrt x }}(0.5)$

$= \dfrac{1}{{2\sqrt {49} }}(0.5) = \dfrac{1}{{14}}(0.5) = 0.035$

$\sqrt {49.5}  \approx 7 + 0.035 = 7.035$

(iii) $\sqrt {0.6} $

$y = \sqrt x $. 

Let $x = 1$ and $\Delta x =  - 0.4$.

$\Delta y = \sqrt {x + \Delta x}  - \sqrt x  = \sqrt {0.6}  - 1$

$\Rightarrow \sqrt {0.6}  = 1 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x$

$= \dfrac{1}{{2\sqrt x }}(\Delta x) = \dfrac{1}{2}( - 0.4)$

$=  - 0.2\sqrt {0.6}  \approx 1 + ( - 0.2) = 1 - 0.2 = 0.8$

(iv) ${(0.009)^{\dfrac{1}{3}}}$

$y = {x^{\dfrac{1}{3}}}$. Let $x = 0.008$ and $\Delta x = 0.001.$

$\Delta y = {(x + \Delta x)^{\dfrac{1}{3}}} - {(x)^{\dfrac{1}{3}}} = {(0.009)^{\dfrac{1}{3}}} - {(0.008)^{\dfrac{1}{3}}}$

$= {(0.009)^{\dfrac{1}{3}}} - 0.2$

$\Rightarrow {(0.009)^{\dfrac{1}{3}}} = 0.2 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{3{{(x)}^{\dfrac{2}{3}}}}}(\Delta x)$

$= \dfrac{1}{{3 \times 0.04}}(0.001) = \dfrac{{0.001}}{{0.12}} = 0.008$

${(0.009)^{\dfrac{1}{3}}}$ is $0.2 + 0.008 = 0.208.$

(v) ${(0.999)^{\dfrac{1}{{10}}}}$

$y = {(x)^{\dfrac{1}{{10}}}}$. 

Let $x = 1$ and $\Delta x =  - 0.001.$

$\Delta y = {(x + \Delta x)^{\dfrac{1}{{10}}}} - {(x)^{\dfrac{1}{{10}}}} = {(0.999)^{\dfrac{1}{{10}}}} - 1$

$\Rightarrow {(0.999)^{\dfrac{1}{{10}}}} = 1 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{10{{(x)}^{\dfrac{9}{{10}}}}}}(\Delta x)$

$= \dfrac{1}{{10}}( - 0.001) =  - 0.0001$

${(0.999)^{\dfrac{1}{{10}}}}$ is $1 + ( - 0.0001) = 0.9999.$

(vi) ${(15)^{\dfrac{1}{4}}}$

$y = {x^{\dfrac{1}{4}}}$. 

Let $x = 16$ and $\Delta x =  - 1$

$\Delta y = {(x + \Delta x)^{\dfrac{1}{4}}} - {x^{\dfrac{1}{4}}} = {(15)^{\dfrac{1}{4}}} - {(16)^{\dfrac{1}{4}}}$

$= {(15)^{\dfrac{1}{4}}} - 2$

$\Rightarrow {(15)^{\dfrac{1}{4}}} = 2 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{4{{(x)}^{\dfrac{3}{4}}}}}(\Delta x)$

$= \dfrac{1}{{4{{(16)}^{\dfrac{3}{4}}}}}( - 1) = \dfrac{{ - 1}}{{4 \times 8}} = \dfrac{{ - 1}}{{32}} =  - 0.03125$

${(15)^{\dfrac{1}{4}}}$ is $2 + ( - 0.03125) = 1.96875.$

(vii) ${(26)^{\dfrac{1}{3}}}$

$y = {(x)^{\dfrac{1}{3}}}$. 

Let $x = 27$ and $\Delta x =  - 1.$

$\Delta y = {(x + \Delta x)^{\dfrac{1}{3}}} - {(x)^{\dfrac{1}{3}}} = {(26)^{\dfrac{1}{3}}} - {(27)^{\dfrac{1}{3}}}$

$= {(26)^{\dfrac{1}{3}}} - 3$

$\Rightarrow {(26)^{\dfrac{1}{3}}} = 3 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{3{{(x)}^{\dfrac{2}{3}}}}}(\Delta x)$

$= \dfrac{1}{{3{{(27)}^{\dfrac{2}{3}}}}}( - 1) = \dfrac{{ - 1}}{{27}} =  - 0.0\overline {370}$

${(26)^{\dfrac{1}{3}}}$ is $3 + ( - 0.0370) = 2.9629.$

(viii) ${(255)^{\dfrac{1}{4}}}$

$y = {(x)^{\dfrac{1}{4}}}.$ 

Let $x = 256$ and $\Delta x =  - 1$

$\Delta y = {(x + \Delta x)^{\dfrac{1}{4}}} - {(x)^{\dfrac{1}{4}}} = {(255)^{\dfrac{1}{4}}} - {(256)^{\dfrac{1}{4}}}$

$= {(255)^{\dfrac{1}{4}}} - 4$

$\Rightarrow {(255)^{\dfrac{1}{4}}} = 4 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{4{{(x)}^{\dfrac{3}{4}}}}}(\Delta x)$

$= \dfrac{1}{{4{{(256)}^{\dfrac{3}{4}}}}}( - 1) = \dfrac{{ - 1}}{{4 \times {4^3}}}$

$=  - 0.0039$

${(255)^{\dfrac{1}{4}}}$ is $4 + ( - 0.0039) = 3.9961.$

(iX) ${(82)^{\dfrac{1}{4}}}$

$y = {x^{\dfrac{1}{4}}}$.

Let $x = 81$ and $\Delta x = 1.$

$\Delta y = {(x + \Delta x)^{\dfrac{1}{4}}} - {(x)^{\dfrac{1}{4}}} = {(82)^{\dfrac{1}{4}}} - {(81)^{\dfrac{1}{4}}}$

$= {(82)^{\dfrac{1}{4}}} - 3$

$\Rightarrow {(82)^{\dfrac{1}{4}}} = \Delta y + 3$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{4{{(x)}^{\dfrac{3}{4}}}}}(\Delta x)$

$= \dfrac{1}{{4{{(81)}^{\dfrac{3}{4}}}}}(1) = \dfrac{1}{{4{{(3)}^3}}}$

$= \dfrac{1}{{108}} = 0.009$

${(82)^{\dfrac{1}{4}}}$ is $3 + 0.009 = 3.009.$

(x) ${(401)^{\dfrac{1}{2}}}$

$y = {x^{\dfrac{1}{2}}}$. Let $x = 400$ and $\Delta x = 1.$

$\Delta y = \sqrt {x + \Delta x}  - \sqrt x  = \sqrt {401}  - \sqrt {400} $

$= \sqrt {401}  - 20$

$\Rightarrow \sqrt {401}  = 20 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{2\sqrt x }}(\Delta x)$

$= \dfrac{1}{{2 \times 20}}(1) = \dfrac{1}{{40}} = 0.025$

$\sqrt {401} $ is $20 + 0.025 = 20.025.$

(xi) ${(0.0037)^{\dfrac{1}{2}}}$

$y = {x^{\dfrac{1}{2}}}.$ Let $x = 0.0036$ and $\Delta x = 0.0001$.

$\Delta y = {(x + \Delta x)^{\dfrac{1}{2}}} - {(x)^{\dfrac{1}{2}}} = {(0.0037)^{\dfrac{1}{2}}} - {(0.0036)^{\dfrac{1}{2}}}$

$= {(0.0037)^{\dfrac{1}{2}}} - 0.06$

$\Rightarrow {(0.0037)^{\dfrac{1}{2}}} = 0.06 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{2\sqrt x }}(\Delta x)$

$= \dfrac{1}{{2 \times 0.06}}(0.0001)$

$= \dfrac{{0.0001}}{{0.12}} = 0.00083$

${(0.0037)^{\dfrac{1}{2}}}$ is $0.06 + 0.00083 = 0.6083.$

(xii) ${(26.57)^{\dfrac{1}{3}}}$

$y = {x^{\dfrac{1}{3}}}$. Let $x = 27$ and $\Delta x =  - 0.43$

$\Delta y = {(x + \Delta x)^{\dfrac{1}{3}}} - {x^{\dfrac{1}{3}}} = {(26.57)^{\dfrac{1}{3}}} - {(27)^{\dfrac{1}{3}}}$

$= {(26.57)^{\dfrac{1}{3}}} - 3$

$26.57{)^{\dfrac{1}{3}}} = 3 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{3{{(x)}^{\dfrac{2}{3}}}}}(\Delta x)$

$= \dfrac{1}{{3(9)}}( - 0.43)$

$= \dfrac{{ - 0.43}}{{27}} =  - 0.015$

${(26.57)^{\dfrac{1}{3}}}$ is $3 + ( - 0.015) = 2.984$

(xiii) ${(81.5)^{\dfrac{1}{4}}}$

$y = {x^{\dfrac{1}{4}}}$. 

Let $x = 81$ and $\Delta x = 0.5$.

$\Delta y = {(x + \Delta x)^{\dfrac{1}{4}}} - {(x)^{\dfrac{1}{4}}} = {(81.5)^{\dfrac{1}{4}}} - {(81)^{\dfrac{1}{4}}}$

$= {(81.5)^{\dfrac{1}{4}}} - 3$

$\Rightarrow {(81.5)^{\dfrac{1}{4}}} = 3 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{4{{(x)}^{\dfrac{3}{4}}}}}(\Delta x)$

$= \dfrac{1}{{4{{(3)}^3}}}(0.5) = \dfrac{{0.5}}{{108}}$

$= 0.0046$

${(81.5)^{\dfrac{1}{4}}}$ is $3 + 0.0046 = 3.0046.$

(xiv) ${(3.968)^{\dfrac{3}{2}}}$

$y = {x^{\dfrac{3}{2}}}$.

Let $x = 4$ and $\Delta x =  - 0.032.$

$\Delta y = {(x + \Delta x)^{\dfrac{3}{2}}} - {x^{\dfrac{3}{2}}} = {(3.968)^{\dfrac{3}{2}}} - {(4)^{\dfrac{3}{2}}}$

$= {(3.968)^{\dfrac{3}{2}}} - 8$

$\Rightarrow {(3.968)^{\dfrac{3}{2}}} = 8 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{3}{2}{(x)^{\dfrac{1}{2}}}(\Delta x)$

$= \dfrac{3}{2}(2)( - 0.032)$

$=  - 0.096$

${(3.968)^{\dfrac{3}{2}}}$ is $8 + ( - 0.096) = 7.904$

$({\mathbf{XV}})$${(32.15)^{\dfrac{1}{{15}}}}$

$y = {x^{\dfrac{1}{5}}}$. Let $x = 32$ and $\Delta x =  - 0.15.$

$\Delta y = {(x + \Delta x)^{\dfrac{1}{5}}} - {x^{\dfrac{1}{5}}} = {(32.15)^{\dfrac{1}{5}}} - {(32)^{\dfrac{1}{5}}} = {(32.15)^{\dfrac{1}{5}}} - 2$

$\Rightarrow {(32.15)^{\dfrac{1}{{15}}}} = 2 + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{5{{(x)}^{\dfrac{4}{5}}}}} \cdot (\Delta x)$

$= \dfrac{1}{{5x{{(2)}^4}}}(0.15) = \dfrac{{0.15}}{{80}} = 0.00187$

${(32.15)^{\dfrac{1}{{15}}}}$ is $2 + 0.00187 = 2.00187.$

2. Find the approximate value of $(2.01)$, where $f(x) = 4{x^2} + 5x + 2$.

Ans: $x = 2$ and $\Delta x = 0.01$

$f(2.01) = f(x + \Delta x) = 4{(x + \Delta x)^2} + 5(x + \Delta x) + 2$

$\Delta y = f(x + \Delta x) - f(x)$

$f(x + \Delta x) = f(x) + \Delta y$

$ \approx f(x) + {f^\prime }(x) \cdot \Delta x$

$\Rightarrow f(2.01) \approx \left( {4{x^2} + 5x + 2} \right) + (8x + 5)\Delta x$

$= \left[ {4{{(2)}^2} + 5(2) + 2} \right] + [8(2) + 5](0.01)$

$= (16 + 10 + 2) + (16 + 5)(0.01)$

$= 28 + (21)(0.01)$

$= 28 + 0.21$

$= 28.21$

$f(2.01)$ is \[28.21.\]

3. Find the approximate value of $f(5.001)$, where $f(x) = {x^3} - 7{x^2} + 15$.

Ans: $x = 5$ and $\Delta x = 0.001$

$f(5.001) = f(x + \Delta x) = {(x + \Delta x)^3} - 7{(x + \Delta x)^2} + 15$

$\Delta y = f(x + \Delta x) - f(x)$

$\therefore f(x + \Delta x) = f(x) + \Delta y$

$ \approx f(x) + {f^\prime }(x) \cdot \Delta x$

$\Rightarrow f(5.001) \approx \left( {{x^3} - 7{x^2} + 15} \right) + \left( {3{x^2} - 14x} \right)\Delta x$

$= \left[ {{{(5)}^3} - 7{{(5)}^2} + 15} \right] + \left[ {3{{(5)}^2} - 14(5)} \right](0.001){\text{ }}$

$= (125 - 175 + 15) + (75 - 70)(0.001)$

$=  - 35 + (5)(0.001)$

$=  - 35 + 0.005$

$=  - 34.995$

$f(5.001)$ is $ - 34.995$

4. Find the approximate change in the volume $V$ of a cube side $x$ meters caused by increasing the side by $1\% $. 

Ans: $V = {x^3}.$

$\therefore dV = \left( {\dfrac{{dV}}{{dx}}} \right)\Delta x$

$= \left( {3{x^2}} \right)\Delta x$

$= \left( {3{x^2}} \right)(0.01x)$

$= 0.03{x^3}$

So, the approx change in the volume of the cube is $0.03{x^3}{m^3}$.

5. Find the approximate change in the surface area of a cube of side $x$ meters caused by decreasing the side by $1\% $. 

Ans: $S = 6{x^2}$

$\therefore \dfrac{{dS}}{{dx}} = \left( {\dfrac{{dS}}{{dx}}} \right)\Delta x$

$= (12x)\Delta x$

$= (12x)(0.01x)$

$= 0.12{x^2}$

So, the approximate change in volume of cube is $0.12{x^2}{m^2}$.

6. If the radius of a sphere is measured as \[7m\]with an error of \[0.02{\text{ }}m,\]then find the approximate error in calculating its volume. 

Ans: $r = 7m$ and $\Delta r = 0.02m$

$V = \dfrac{4}{3}\pi {r^3}$

$\therefore \dfrac{{dV}}{{dr}} = 4\pi {r^2}$

$\therefore dV = \left( {\dfrac{{dV}}{{dr}}} \right)\Delta r$

$\quad  = \left( {4\pi {r^2}} \right)\Delta r$

$= 4\pi {(7)^2}(0.02){m^3}$

$= 3.92\pi {m^3}$

So, the approx error in calculating the volume is $3.92\pi {{\text{m}}^3}$.

7. If the radius of a sphere is measured as \[9m\]with an error of\[0.03m\], then find the approximate error in calculating in surface area. 

Ans: $r = 9m$ and $\Delta r = 0.03m$

$\therefore \dfrac{{dS}}{{dr}} = \dfrac{d}{{dr}}\left( {4\pi {r^2}} \right) = 8\pi r$

$\therefore dS = \left( {\dfrac{{dS}}{{dr}}} \right)\Delta r$

$= (8\pi r)\Delta r$

$= 8\pi (9)(0.03){m^2}$

$= 2.16\pi {m^2}$

So, the approx error in calculating the surface area is $2.16\pi {m^2}$.

8. If $f(x) = 3{x^2} + 15x + 5$, then the approximate value of $(3.02)$ is

A. 47.66, B.57.66, C. 67.66, D. \[77.66\]

Ans: $f(3.02) = f(x + \Delta x) = 3{(x + \Delta x)^2} + 15(x + \Delta x) + 5$

$\Delta y = f(x + \Delta x) - f(x)$

$\Rightarrow f(x + \Delta x) = f(x) + \Delta y$

$ \approx f(x) + {f^\prime }(x)\Delta x$

$\Rightarrow f(3.02) \approx \left( {3{x^2} + 15x + 5} \right) + (6x + 15)\Delta x$

$= \left[ {3\left( {{3^2}} \right) + 15(3) + 5} \right] + [6(3) + 15](0.02)$

$= (27 + 45 + 5) + (18 + 15)(0.02)$

$= 77 + (33)(0.02)$

$= 77 + 0.66$

$= 77.66$

So, approx value of $(3.02)$ is \[77.66.\]

The correct answer is ${\mathbf{D}}$.

9. The approximate change in the volume of a cube of side $x$ meters caused by increasing the side by $3\% $ is

Ans: $V = {x^3}$

$\therefore dV = \left( {\dfrac{{dV}}{{dx}}} \right)\Delta x$

$= \left( {3{x^2}} \right)\Delta x$

$= \left( {3{x^2}} \right)(0.03x)$

$= 0.09{x^3}{m^3}$

So, the approx change in the volume of the cube is $0.09{x^3}{m^3}$. 

The correct answer is ${\mathbf{C}}$.

Exercise 6.5

1. Find the maximum and minimum values, if any, of the following given by

(i) $f(x) = {(2x - 1)^2} + 3$

(ii) $f(x) = 9{x^2} + 12x + 2$

(iii) $f(x) =  - {(x - 1)^2} + 10$

(iv)$g(x) = {x^3} + 1$

Ans: (i) $f(x) = {(2x - 1)^2} + 3$

${(2x - 1)^2} \geqslant 0$ for every $x \in {\mathbf{R}}$.

$f(x) = {(2x - 1)^2} + 3 \geqslant 3$ for $x \in {\mathbf{R}}$.

The minimum value of $f$ occurs when when $2x - 1 = 0$.

$2x - 1 = 0,x = \dfrac{1}{2}$

Min value of $f\left( {\dfrac{1}{2}} \right) = {\left( {2 \cdot \dfrac{1}{2} - 1} \right)^2} + 3 = 3$.

The function $f$ does not have a maximum value.

(ii) $f(x) = 9{x^2} + 12x + 2 = {\left( {3{x^2} + 2} \right)^2} - 2$.

${\left( {3{x^2} + 2} \right)^2} \geqslant 0$ for $x \in {\mathbf{R}}$.

$f(x) = {\left( {3{x^2} + 2} \right)^2} - 2 \geqslant  - 2$ for $x \in {\mathbf{R}}$.

minimum value of $f$ is when $3x + 2 = 0$. 

$3x + 2 = 0 = 0,$

$x = \dfrac{{ - 2}}{3}$

Minimum value of $f\left( { - \dfrac{2}{3}} \right) = {\left( {3\left( {\dfrac{{ - 2}}{3}} \right) + 2} \right)^2} - 2 =  - 2$

$f$ does not have a maximum value.

(iii) $f(x) =  - {(x - 1)^2} + 10$

${(x - 1)^2} \geqslant 0$ for $x \in {\mathbf{R}}$.

$f(x) =  - {(x - 1)^2} + 10 \leqslant 10$ for $x \in {\mathbf{R}}$.

maximum value of $f$ is when $(x - 1) = 0$.

$(x - 1) = 0,x = 0$

Maximum value of $f = f(1) =  - {(1 - 1)^2} + 10 = 10$

$f$ does not have a minimum value.

(iv) $g(x) = {x^3} + 1.$

$g$ neither has a maximum value nor a minimum value.

2. Find the maximum and minimum values, if any, of the following functions given by

(i) $f(x) = |x + 2| - 1$

(ii) $g(x) =  - |x + 1| + 3$

(iii) $h(x) = \sin (2x) + 5$

(iv) $f(x) = |\sin 4x + 3|$

(v) $h(x) = x + 4,x \in ( - 1,1)$

Ans:  (i) $f(x) = |x + 2| - 1$

$|x + 2| \geqslant 0$ for $x \in {\mathbf{R}}$.

$f(x) = |x + 2| - 1 \geqslant  - 1$ for $x \in {\mathbf{R}}$.

minimum value of $f$ is when $|x + 2| = 0$.

 $|x + 2| = 0$

$\Rightarrow x =  - 2$

Minimum value of $f = f( - 2) = | - 2 + 2| - 1 =  - 1$

 $f$ does not have a maximum value.

(ii) $g(x) =  - |x + 1| + 3$

$ - |x + 1| \leqslant 0$ for $x \in {\mathbf{R}}$.

$g(x) =  - |x + 1| + 3 \leqslant 3$ for $x \in {\mathbf{R}}$.

maximum value of $g$ is when $|x + 1| = 0$.

 $|x + 1| = 0$

$\Rightarrow x =  - 1$

Maximum value of $g = g( - 1) =  - | - 1 + 1| + 3 = 3$

g does not have a minimum value.

(iii) $h(x) = \sin 2x + 5$

$- 1 \leqslant \sin 2x \leqslant 1$

$- 1 + 5 \leqslant \sin 2x + 5 \leqslant 1 + 5$

$4 \leqslant \sin 2x + 5 \leqslant 6$

maximum and minimum values of $h$ are 6 and 4 respectively.

(iv) $f(x) = |\sin 4x + 3|$

$ - 1 \leqslant \sin 4x \leqslant 1$

$2 \leqslant \sin 4x + 3 \leqslant 4$

$2 \leqslant |\sin 4x + 3| \leqslant 4$

maximum and minimum values of $f$ are 4 and 2 respectively.

(v) $h(x) = x + 4,x \in ( - 1,1)$

Here, if a point ${x_0}$ is closest to $ - 1$, then we find $\dfrac{{{x_0}}}{2} + 1 < {x_0} + 1$ for all ${x_0} \in ( - 1,1)$.

Also, if ${x_1}$ is closet to $ - 1$, then we find ${x_1} + 1 < \dfrac{{{x_1} + 1}}{2} + 1$ for all ${x_0} \in ( - 1,1)$. 

function has neither maximum nor minimum value in $( - 1,1)$.

3. Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

(i) $f(x) = {x^2}$

(ii) $g(x) = {x^3} - 3x$

(iii) $h(x) = \sin x + \cos .0 < x < \dfrac{\pi }{2}$

(iv) $f(x) = \sin x - \cos x,0 < x < 2\pi $ 

(v) $f(x) = {x^3} - 6{x^2} + 9x + 15$

(vi) $g(x) = \dfrac{x}{2} + \dfrac{2}{x},x > 0$

(vii) $g(x) = \dfrac{1}{{{x^2} + 2}}$

(vii) $f(x) = x\sqrt {1 - x} ,x > 0$

Ans: (i) $f(x) = {x^2}$

$\therefore {f^\prime }(x) = 2x$

${f^\prime }(x) = 0 \Rightarrow x = 0$

We have ${f^\prime }(0) = 2$,

by second derivative test, $x = 0$ is a point of local minima and local minimum value of $f$

at $x = 0$ is $f(0) = 0$.

(ii) $g(x) = {x^3} - 3x$

$\therefore {g^\prime }(x) = 3{x^2} - 3$

${g^\prime }(x) = 0 \Rightarrow 3{x^2} = 3$

$\Rightarrow x =  \pm 1$

${g^\prime }(x) = 6x$

${g^\prime }(1) = 6 > 0$

${g^\prime }( - 1) =  - 6 < 0$

By second derivative test, $x = 1$ is a point of local minima and local minimum value of $g$

At $x = 1$ is $g(1) = {1^3} - 3 = 1 - 3 =  - 2$.

$x =  - 1$ is a point of local maxima and local maximum value of $g$ at

$x =  - 1$ is $g(1) = {( - 1)^3} - 3( - 1) =  - 1 + 3 =  - 2$.

(iii) $h(x) = \sin x + \cos .0 < x < \dfrac{\pi }{2}$

$\therefore {h^\prime }(x) = \cos x + \sin x$

${h^\prime }(x) = 0 \Rightarrow \sin x = \cos x$

$\Rightarrow \tan x = 1$

$\Rightarrow x = \dfrac{\pi }{4} \in \left( {0,\dfrac{\pi }{2}} \right)$

${h^\prime }(x) = \sin x - \cos x =  - (\sin x + \cos x)$

$h\left( {\dfrac{\pi }{4}} \right) =  - \left( {\dfrac{1}{{\sqrt 2 }} + \dfrac{1}{{\sqrt 2 }}} \right) =  - \dfrac{2}{{\sqrt 2 }} =  - \sqrt 2  < 0$

Therefore, by second derivative test, $x = \dfrac{\pi }{4}$ is a point of local maxima and the local Maximum value of $h$ at $x = \dfrac{\pi }{4}$ is $h\left( {\dfrac{\pi }{4}} \right) = \sin \dfrac{\pi }{4} + \cos \dfrac{\pi }{4}$

$= \dfrac{1}{{\sqrt 2 }} + \dfrac{1}{{\sqrt 2 }} = \sqrt 2 $

(iv) $f(x) = \sin x - \cos x,0 < x < 2\pi $

$\therefore {f^\prime }(x) = \cos x + \sin x$

${f^\prime }(x) = 0 \Rightarrow \cos x =  - \sin x \Rightarrow \tan x =  - 1$

$\Rightarrow x = \dfrac{{3\pi }}{4},\dfrac{{7\pi }}{4} \in (0,2\pi )$

${f^{\prime \prime }}(x) =  - \sin x + \cos x$

$=  - \dfrac{1}{{\sqrt 2 }} - \dfrac{1}{{\sqrt 2 }} =  - \sqrt 2  > 0$

$= \dfrac{1}{{\sqrt 2 }} + \dfrac{1}{{\sqrt 2 }} = \sqrt 2  > 0$

by second derivative test, $x = \dfrac{{3\pi }}{4}$ is a point of local maxima and the local maximum value of

$f$ at $x = \dfrac{{3\pi }}{4}$ is

$f\left( {\dfrac{{3\pi }}{4}} \right) = \sin \dfrac{{3\pi }}{4}\cos \dfrac{{3\pi }}{4}$

$= \dfrac{1}{{\sqrt 2 }} + \dfrac{1}{{\sqrt 2 }} = \sqrt 2 $

$x = \dfrac{{7\pi }}{4}$ is a point of local minima and the local minimum value of $f$ at $x = \dfrac{{7\pi }}{4}$ is $f\left( {\dfrac{{7\pi }}{4}} \right) = \sin \dfrac{{7\pi }}{4} - \cos \dfrac{{7\pi }}{4}$

$=  - \dfrac{1}{{\sqrt 2 }} - \dfrac{1}{{\sqrt 2 }} =  - \sqrt 2 $

(v) $f(x) = {x^3} - 6x + 9x + 15$

$\therefore {f^\prime }(x) = 3{x^2} - 12x + 9$

$f(x) = 0 \Rightarrow 3\left( {{x^2} - 4x + 3} \right) = 0$

$\Rightarrow 3(x - 1)(x - 3) = 0$

$\Rightarrow x = 1,3$

${f^{\prime \prime }}(x) = 6x - 12 = 6(x - 2)$

${f^{\prime \prime }}(1) = 6(1 - 2) =  - 6 < 0$

${f^{\prime \prime }}(3) = 6(3 - 2) = 6 > 0$

by second derivative test, $x = 1$ is a point of local maxima and the local maximum value of $f$ at

$x = 1$ is $f(1) = 1 - 6 + 9 + 15 = 19$.

$x = 3$ is a point of local minima and the local minimum value of $f$ at $x = 3$ is

$f(3) = 27 - 54 + 27 + 15 = 15$.

(vi) $g(x) = \dfrac{x}{2} + \dfrac{2}{x},x > 0$

$\therefore {g^\prime }(x) = \dfrac{1}{2} - \dfrac{2}{{{x^2}}}$

${g^\prime }(x) = 0 \Rightarrow \dfrac{2}{{{x^2}}} = \dfrac{1}{2} \Rightarrow {x^3} = 4$

$x > 0,x = 2$.

${g^{\prime \prime }}(x) = \dfrac{4}{{{x^3}}}$

${g^{\prime \prime }}(2) = \dfrac{4}{{{2^3}}} = \dfrac{1}{2} > 0$

by second derivative test, $x = 2$ is a point of local minima and the local minimum value of $g$ at

$x = 2$ is $g(2) = \dfrac{2}{2} + \dfrac{2}{2} = 1 + 1 = 2.$

(vii) $g(x) = \dfrac{1}{{{x^2} + 2}}$

$\therefore {g^\prime }(x) = \dfrac{{ - (2x)}}{{{{\left( {{x^3} + 2} \right)}^2}}}$

${g^\prime }(x) = 0 \Rightarrow \dfrac{{ - 2x}}{{{{\left( {{x^3} + 2} \right)}^2}}} = 0$

$\Rightarrow x = 0$

for values close to $x = 0$ and left of $0,{g^1}(x) > 0$ for values close to $x = 0$ and to right of $0{g^1}(x) < 0$

by first derivative test $x = 0$ is a point of local maxima and the local maximum value of $g(0)$ is $\dfrac{1}{{0 + 2}} = \dfrac{1}{2}$

(viii) $f(x) = x\sqrt {1 - x} ,x > 0$

$\therefore {f^\prime }(x) = x\sqrt {1 - x}  + x \cdot \dfrac{1}{{2\sqrt {1 - x} }}( - 1) = \sqrt {1 - x}  - \dfrac{x}{{2\sqrt {1 - x} }}$

$= \dfrac{{2(1 - x) - x}}{{2\sqrt {1 - x} }} - \dfrac{{2 - 3x}}{{2\sqrt {1 - x} }}$

${f^\prime }(x) = 0 \Rightarrow \dfrac{{2 - 3x}}{{2\sqrt {1 - x} }} = 0 \Rightarrow 2 - 3x = 0$

$\Rightarrow x = \dfrac{2}{3}$

${f^{\prime \prime }}(x) = \dfrac{1}{2}\left[ {\dfrac{{\sqrt {1 - x} ( - 3) - (2 - 3x)\left( {\dfrac{{ - 1}}{{2\sqrt {1 - x} }}} \right)}}{{1 - x}}} \right]$

$= \dfrac{{\sqrt {1 - x} ( - 3) + 2(2 - 3x)\left( {\dfrac{1}{{2\sqrt {1 - x} }}} \right)}}{{2(1 - x)}}$

$= \dfrac{{ - 6(1 - x) + 2(2 - 3x)}}{{4{{(1 - x)}^{\dfrac{3}{2}}}}}$

$= \dfrac{{3x - 4}}{{4{{(1 - x)}^{\dfrac{3}{2}}}}}$

${f^{\prime \prime }}\left( {\dfrac{2}{3}} \right) = \dfrac{{3\left( {\dfrac{2}{3}} \right) - 4}}{{4{{\left( {1 - \dfrac{2}{3}} \right)}^{\dfrac{3}{2}}}}} = \dfrac{{2 - 4}}{{4{{\left( {\dfrac{1}{3}} \right)}^{\dfrac{3}{2}}}}} = \dfrac{{ - 1}}{{2{{\left( {\dfrac{1}{3}} \right)}^{\dfrac{3}{2}}}}} < 0$

by second derivative test, $x = \dfrac{2}{3}$ is a point of local maxima and the local maximum value of $f$

$x = \dfrac{2}{3}$

$f\left( {\dfrac{2}{3}} \right) = \dfrac{2}{3}\sqrt {1 - \dfrac{2}{3}}  = \dfrac{2}{3}\sqrt {\dfrac{1}{3}}  = \dfrac{2}{{3\sqrt 3 }} = \dfrac{{2\sqrt 3 }}{9}$

4. Prove that the following functions do not have maxima or minima:

(i) $f(x) = {e^x}$

(ii) $g(x) = \log x$

(iii) $h(x) = {x^3} + {x^2} + x + 1$

Ans: (i) $f(x) = {e^x}$

$\therefore {f^\prime }(x) = {e^x}$

if ${f^\prime }(x) = 0,{e^x} = 0.$ But exponential function can never be 0 for any value of $x$.

There is no $c \in {\mathbf{R}}$ such that ${f^\prime }(c) = 0$.

$f$ does not have maxima or minima.

(ii) We have, $g(x) = \log x$

$\therefore {g^\prime }(x) = \dfrac{1}{x}$

$\log x$ is defined for positive $x,{g^\prime }(x) > 0$ for any $x$.

there does not exist $c \in {\mathbf{R}}$ such that ${g^\prime }(c) = 0$

function $g$ does not have maxima or minima.

(iii) We have, $h(x) = {x^3} + {x^2} + x + 1$

$\therefore {h^\prime }(x) = 3{x^2} + 2x + 1$

there does not exist $c \in {\mathbf{R}}$ such that ${h^\prime }(c) = 0$. function $h$ does not have maxima or minima.

5. Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

(i) $f(x) = {x^3},x \in [ - 2,2]$

(iii) $f(x) = 4x - \dfrac{1}{2}{x^2},x \in \left[ { - 2,\dfrac{9}{2}} \right]$

(ii) $f(x) = \sin x + \cos x,x \in [0,\pi ]$

(iv) $f(x) = {(x - 1)^2} + 3,x \in [ - 3,1]$

Ans: (i) $f(x) = {x^3}$. $\therefore {f^\prime }(x) = 3{x^2}$

${f^\prime }(x) = 0 \Rightarrow x = 0$

$f(0) = 0$

$f( - 2) = {( - 2)^3} =  - 8$

$f(2) = {(2)^3} = 8$

Hence, the absolute maximum of $f$ on $[ - 2,2]$.is 8 at $x =  - 2$.

the absolute minimum of $f$ on $[ - 2,2]$ is $ - 8$ at $x =  - 2$.

(ii) $f(x) = \sin x + \cos x.$

$\therefore {f^\prime }(x) = \cos x - \sin x$

${f^\prime }(x) = 0 \Rightarrow \sin x = \cos x$

$\Rightarrow \tan x = 1 \Rightarrow x = \dfrac{\pi }{4}$

$f\left( {\dfrac{\pi }{4}} \right) = \sin \dfrac{\pi }{4} + \cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }} + \dfrac{1}{{\sqrt 2 }} = \dfrac{2}{{\sqrt 2 }} = \sqrt 2 $

$f(0) = \sin 0 + \cos 0 = 0 + 1 = 1$

$f(\pi ) = \sin \pi  + \cos \pi  = 0 - 1 = 1$

the absolute maximum of $f$ on $[0,\pi ]$ is $\sqrt 2 $ at $x = \dfrac{\pi }{4}$

the absolute minimum of $f$ on $[0,\pi ]$ is $ - 1$ at $x = \pi $.

(iii) $f(x) = 4x - \dfrac{1}{2}{x^2}$

$\therefore {f^\prime }(x) = 4x - \dfrac{1}{2}(2x) = 4 - x$

${f^\prime }(x) = 0 \Rightarrow x = 4$

$f(4) = 16 - \dfrac{1}{2}(16) = 16 - 8 = 8$

$f( - 2) =  - 8 - \dfrac{1}{2}(4) =  - 8 - 2 =  - 10$

$f\left( {\dfrac{9}{2}} \right) = 4\left( {\dfrac{9}{2}} \right) - \dfrac{1}{2}{\left( {\dfrac{9}{2}} \right)^2} = 18 - \dfrac{{81}}{8}$

$= 18 - 10.125 = 7.875$

the absolute maximum of $f$ on $\left[ { - 2,\dfrac{9}{2}} \right]$ is 8 at $x = 4$

the absolute minimum of $f$ on $\left[ { - 2,\dfrac{9}{2}} \right]$ is $ - 10$ at $x =  - 2$.

(iv) $f(x) = {(x - 1)^2} + 3$

$\therefore {f^\prime }(x) = 2(x - 1)$

${f^\prime }(x) = 0 \Rightarrow 2(x - 1) = 0,x = 1$

$f(1) = {(1 - 1)^2} + 3 = 0 + 3 = 3$

$f( - 3) = {( - 3 - 1)^2} + 3 = 16 + 3 = 19$

absolute maximum value of $f$ on $[ - 3,1]$ is 19 at $x =  - 3$

minimum value of $f$ on $[ - 3,1]$ is at $x = 1$.

6. Find the maximum profit that a company can make, if the profit function is given by $p(x) = 41 - 24x - 18{x^2}$

Ans: $p(x) = 41 - 24x - 18{x^2}.$

$\therefore {p^\prime }(x) =  - 24 - 36x$

${p^{\prime \prime }}(x) =  - 36$

${p^\prime }(x) = 0 \Rightarrow x = \dfrac{{ - 24}}{{36}} =  - \dfrac{2}{3}$

${p^{\prime \prime }}\left( {\dfrac{{ - 2}}{3}} \right) =  - 36 < 0$

By second derivative test, $x =  - \dfrac{2}{3}$ is the point of local maximum of $p$.

$\therefore $ Maximum profit $= p = \left( { - \dfrac{2}{3}} \right)$

$= 41 - 24\left( { - \dfrac{2}{3}} \right) - 18{\left( { - \dfrac{2}{3}} \right)^2} = 41 + 16 - 8$

7. Find the intervals in which the function $f$ given by $f(x) = {x^3} + \dfrac{1}{{{x^3}}},x \ne 0$ is

(i) Increasing

(ii) Decreasing 

Ans: $f(x) = {x^3} + \dfrac{1}{{{x^3}}}$

$\therefore {f^\prime }(x) = 3{x^2} - \dfrac{3}{{{x^4}}} = \dfrac{{3{x^6} - 3}}{{{x^4}}}$

${f^\prime }(x) = 0 \Rightarrow 3{x^6} - 3 = 0$

$\Rightarrow {x^6} = 1 \Rightarrow x =  \pm 1$

In $( - \infty , - 1)$ and $(1,\infty ),{f^\prime }(x) > 0.$

when $x <  - 1$ and $x > 1,f$ is increasing.

In $( - 1,1),{f^\prime }(x) < 0$.

when $ - 1 < x < 1,f$ is decreasing.

8. At what points in the interval $[0,2\pi ]$, does the function $\sin 2x$ attain, its maximum value? 

Ans: $f(x) = \sin 2x$.

$\therefore {f^\prime }(x) = 2\cos 2x$

${f^\prime }(x) = 0 \Rightarrow \cos 2x = 0$

$\Rightarrow 2x = \dfrac{\pi }{2},\dfrac{{3\pi }}{2},\dfrac{{5\pi }}{2},\dfrac{{7\pi }}{2}$

$\Rightarrow x = \dfrac{\pi }{4},\dfrac{{3\pi }}{4},\dfrac{{5\pi }}{4},\dfrac{{7\pi }}{4}$

$f\left( {\dfrac{\pi }{4}} \right) = \sin \dfrac{\pi }{2} = 1,$$f\left( {\dfrac{{3\pi }}{4}} \right) = \sin \dfrac{{3\pi }}{2} =  - 1$

$f\left( {\dfrac{{5\pi }}{4}} \right) = \sin \dfrac{{5\pi }}{2} = 1,$$f\left( {\dfrac{{7\pi }}{4}} \right) = \sin \dfrac{{7\pi }}{2} =  - 1$

$f(0) = \sin 0 = 0,f(2\pi ) = \sin 2\pi  = 0$

absolute maximum value of $f[0,2\pi ]$ is at $x = \dfrac{\pi }{4}$ and $x = \dfrac{{5\pi }}{4}$.

9. What is the maximum value of the function $\sin x + \cos x$ ?

Ans: $f(x) = \sin x + \cos x$

$\therefore {f^\prime }(x) = \cos x - \sin x$

${f^\prime }(x) = 0 \Rightarrow \sin x = \cos x \Rightarrow \tan x = 1 \Rightarrow x = \dfrac{\pi }{4},\dfrac{{5\pi }}{4} \ldots $

${f^\prime }(x) =  - \sin x - \cos x =  - (\sin x + \cos x)$

${f^{\prime \prime }}(x)$ will be negative when $(\sin x + \cos x)$ is positive

we know that $\sin x$ and $\cos x$ are positive in the first quadrant ${f^{\prime \prime }}(x)$ will be negative when $x \in \left( {0,\dfrac{\pi }{2}} \right)$.

consider $x = \dfrac{\pi }{4}$. ${f^{\prime \prime }}\left( {\dfrac{\pi }{4}} \right) =  - \left( {\sin \dfrac{\pi }{4} + \cos \dfrac{\pi }{4}} \right) =  - \left( {\dfrac{2}{{\sqrt 2 }}} \right) =  - \sqrt 2  < 0$

By second derivative test, $f$ will be the maximum at $x = \dfrac{\pi }{4}$ and the maximum value of $f$ is

$f\left( {\dfrac{\pi }{4}} \right) = \sin \dfrac{\pi }{4} + \cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{{\sqrt 2 }} = \dfrac{2}{{\sqrt 2 }} = \sqrt 2 $

10. Find the maximum value of $2{x^3} - 24x + 107$ in the interval $[1,3]$. Find the maximum value of the same function in $[ - 3, - 1]$.

Ans: $f(x) = 2{x^3} - 24x + 107$

$\therefore {f^\prime }(x) = 6{x^2} - 24 = 6\left( {{x^2} - 4} \right)$

${f^\prime }(x) = 0 \Rightarrow 6\left( {{x^2} - 4} \right) = 0 \Rightarrow {x^2} = 4 \Rightarrow x =  \pm 2$

consider\[[1,3]\]. 

$f(2) = 2(8) - 24(2) + 107 = 16 - 48 + 107 = 75$

$f(1) = 2(1) - 24(1) + 107 = 2 - 24 + 107 = 85$

$f(3) = 2(27) - 24(3) + 107 = 54 - 72 + 107 = 89$

absolute maximum of $f(x)$ in the \[[1,3]\] is 89 at $x = 3$.

consider $[ - 3, - 1]$. 

$f( - 3) = 2( - 27) - 24( - 3) + 107 = 54 + 72 + 107 = 125$

$f( - 1) = 2( - 1) - 24( - 1) + 107 = 2 + 24 + 107 = 129$

$f( - 2) = 2( - 8) - 24( - 2) + 107 =  - 16 + 48 + 107 = 139$

absolute maximum of $f(x)$ in $[ - 3, - 1]$ is 139 at $x =  - 2$.

11. It is given that at $x = 1$, the function ${x^4} - 62{x^2} + ax + 9$ attains its maximum value, on the interval $[0,2]$. Find the value of $a$.

Ans: $f(x) = {x^4} - 62{x^2} + ax + 9.$

$\therefore {f^\prime }(x) = 4{x^2} - 124x + a$

$\therefore {f^\prime }(1) = 0$

$\Rightarrow 4 - 124 + a = 0$

$\Rightarrow a = 120$

the value of $a$ is 120.

12. Find the maximum and minimum values of $x + \sin 2x$ on $[0,2\pi ]$.

Ans: $f(x) = x + \sin 2x$

$\therefore {f^\prime }(x) = 1 + 2\cos 2x$

${f^\prime }(x) = 0 \Rightarrow \cos 2x =  - \dfrac{1}{2} =  - \cos \dfrac{\pi }{3} = \cos \left( {\pi  - \dfrac{\pi }{3}} \right) = \cos \dfrac{{2\pi }}{3}$

$2x = 2\pi  \pm \dfrac{{2\pi }}{{{3^\prime }}}n \in {\mathbf{Z}}$

$\Rightarrow x = \pi  \pm \dfrac{\pi }{3},n \in {\mathbf{Z}}$

$\Rightarrow x = \dfrac{\pi }{3},\dfrac{{2\pi }}{3},\dfrac{{4\pi }}{3},\dfrac{{5\pi }}{3} \in [0,2\pi ]$

$f\left( {\dfrac{\pi }{3}} \right) = \dfrac{\pi }{3} + \sin \dfrac{{2\pi }}{3} = \dfrac{\pi }{3} + \dfrac{{\sqrt 3 }}{2}$

$f\left( {\dfrac{{2\pi }}{3}} \right) = \dfrac{{2\pi }}{3} + \sin \dfrac{{4\pi }}{3} = \dfrac{{2\pi }}{3} - \dfrac{{\sqrt 3 }}{2}$

$f\left( {\dfrac{{4\pi }}{3}} \right) = \dfrac{{4\pi }}{3} + \sin \dfrac{{8\pi }}{3} = \dfrac{{4\pi }}{3} + \dfrac{{\sqrt 3 }}{2}$

$f\left( {\dfrac{{5\pi }}{3}} \right) = \dfrac{{5\pi }}{3} + \sin \dfrac{{10\pi }}{3} = \dfrac{{5\pi }}{3} - \dfrac{{\sqrt 3 }}{2}$

$f(0) = 0 + \sin 0 = 0$

$f(2\pi ) = 2\pi  + \sin 4\pi  = 2\pi  + 0 = 2\pi $

absolute maximum value of $f(x)$ in $[0,2\pi ]$ is $2\pi $ at $x = 2\pi $ 

absolute minimum value of $f(x)$ in $[0,2\pi ]$ is 0 at $x = 0$.

13. Find two numbers whose sum is 24 and whose product is as large as possible. 

Ans: Let number be $x$. The other number is $(24 - x)$. $p(x)$ denote the product of the two numbers.

$P(x) = x(24 - x) = 24x - {x^2}$

$\therefore {P^\prime }(x) = 24 - 2x$

${P^{\prime \prime }}(x) =  - 2$

${P^\prime }(x) = 0 \Rightarrow x = 12$

${P^{\prime \prime }}(12) =  - 2 < 0$

$x = 12$ is point of local maxima of $P$. 

Product of the numbers is the maximum when numbers are 12 and $24 - 12 = 12$.

14. Find two positive numbers $x$ and $y$ such that $x + y = 60$ and $x{y^3}$ is maximum. 

Ans: numbers are $x$ and $y$ such that $x + y = 60$. $y = 60 - x$

$f(x) = x{y^3}$

$\Rightarrow f(x) = x{(60 - x)^3}$

$\therefore {f^\prime }(x) = {(60 + x)^3} - 3x{(60 - x)^2}$

$= {(60 + x)^3}[60 - x - 3x]$

$= {(60 + x)^3}(60 - 4x)$

${f^{\prime \prime }}(x) =  - 2(60 - x)(60 - 4x) - 4{(60 - x)^2}$

$=  - 2(60 - x)[60 - 4x + 2(60 - x)]$

$=  - 2(60 - x)(180 - 6x)$

$=  - 12(60 - x)(30 - x)$

${f^\prime }(x) = 0 \Rightarrow x = 60$ or $x = 15$

$x = 60,{f^{\prime \prime }}(x) = 0$

$x = 15,$${f^{\prime \prime }}(x) =  - 12(60 - 15)(30 - 15) = 12 \times 45 \times 15 < 0$

$x = 15$ is a point of local maxima of $f$.

function $x{y^3}$ is maximum when $x = 15$ and $y = 60 - 15 = 45$.

Required numbers are 15 and 45.

15. Find two positive numbers $x$ and $y$ such that their sum is 35 and the product ${x^2}{y^5}$ is a maximum

Ans: one number be $x$. other number is $y = (35 - x)$.

$p(x) = {x^2}{y^5}$

$P(x) = {x^2}{(35 - x)^5}$

$\therefore {P^\prime }(x) = 2x{(35 - x)^5} - 5{x^2}{(35 - x)^4}$

$= x{(35 - x)^4}[2(35 - x) - 5x]$

$= x{(35 - x)^4}(70 - 7x)$

$= 7x{(35 - x)^4}(10 - x)$

And, ${P^{\prime \prime }}(x) = 7{(35 - x)^4}(10 - x) + 7x\left[ { - {{(35 - 5)}^4} - 4{{(35 - x)}^3}(10 - x)} \right]$

$= 7{(35 - x)^4}(10 - x) - 7x{(35 - x)^4} - 28x{(35 - x)^3}(10 - x)$

$= 7{(35 - x)^3}[(35 - x)(10 - x) - x(35 - x) - 4x(10 - x)]$

$= 7{(35 - x)^3}\left[ {350 - 45x + {x^2} - 35x + {x^2} - 40x + 4{x^2}} \right]$

$= 7{(35 - x)^3}\left( {6{x^2} - 120x + 350} \right)$

${P^\prime }(x) = 0 \Rightarrow x = 0,x = 35,x = 10$

$x = 35,{f^\prime }(x) = f(x) = 0$ and $y = 35 - 35 = 0.$

$x = 0,y = 35 - 0 = 35$ and product ${x^2}{y^2}$ will be 0 .

 $x = 0$ and $x = 35$ cannot be the possible values of $x$. 

$x = 10$, 

${P^{\prime \prime }}(x) = 7{(35 - 10)^3}(6 \times 100 - 120 \times 10 + 350)$

$= 7{(25)^3}( - 250) < 0$

$P(x)$ will be the maximum when $x = 10$ and $y = 35 - 10 = 25$. the numbers are 10 and 25.

16. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Ans: one number be $x$. the other number is $(16 - x)$. sum of cubes of these numbers be denoted by $S(x)$. $S(x) = {x^3} + {(16 - x)^3}$

$\therefore {S^\prime }(x) = 3{x^2} - 3{(16 - x)^2},$

${S^{\prime \prime }}(x) = 6x + 6(16 - x)$

${S^\prime }(x) = 0 \Rightarrow 3{x^2} - 3{(16 - x)^2} = 0$

$\Rightarrow {x^2} - {(16 - x)^2} = 0$

$\Rightarrow {x^2} - 256 - {x^2} + 32x = 0$

$\Rightarrow x = \dfrac{{256}}{{32}} = 8$

${S^{\prime \prime }}(8) = 6(8) + 6(16 - 8) = 48 + 48 = 96 > 0$

By second derivative test, $x = 8$ is point of local minima of $S$. sum of the cubes of the numbers is minimum when the numbers are 8 and $16 - 8 = 8$.

17. A square piece of tin od side $18\;{\text{cm}}$ is to made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible? 

Ans: side of the square to be cut off be $x\;{\text{cm}}$.

The length and breath of the box will be $(18 - 2x){\text{cm}}$ each and the height of the box is $x\;{\text{cm}}$.

$V(x) = x{(18 - 2x)^2}$

$\therefore {V^\prime }(x) = {(18 - 2x)^2} - 4x(18 - 2x)$

$= (18 - 2x)[18 - 2x - 4x]$

$= (18 - 2x)(18 - 6x)$

$= 6 \times 2(9 - x)(3 - x)$

$= 12(9 - x)(3 - x)$

${V^{\prime \prime }}(x) = 12[ - (9 - x) - (3 - x)]$

$=  - 12(9 - x + 3 - x)$

$=  - 12(12 - 2x)$

$=  - 24(6 - x)$

${v^\prime }(x) = 0 \Rightarrow x = 9$ or $x = 3$

$x = 9$, then the length and the breadth will become $0 .$

$\therefore x \ne 9$.

$\Rightarrow x = 3$.

${V^{\prime \prime }}(3) =  - 24(6 - 3) =  - 72 < 0$

$\therefore $ By second derivative test, $x = 3$ is the point of maxima of $V$.

18. A rectangular sheet of tin $45\;{\text{cm}}$ by $24\;{\text{cm}}$ is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible? 

Ans: side of the square to be cut be $x\;{\text{cm}}$.

height of the box is $x$, the length is $45 - 2x$,

breadth is $24 - 2x$.

$V(x) = x(45 - 2x)(24 - 2x)$

$= x\left( {1080 - 90x - 48x + 4{x^2}} \right)$

$= 4{x^3} - 138{x^2} + 1080x$

\[\therefore V'(x) = 12{x^2} - 276 + 1080\]

\[ = 12\left( {{x^2} - 23x + 90} \right)\]

$= 12(x - 18)(x - 5)$ 

${V^{\prime \prime }}(x) = 24x - 276 = 12(2x - 23)$

${V^\prime }(x) = 0 \Rightarrow x = 18$ and $x = 5$

not possible to cut a square of side $18\;{\text{cm}}$ from each corner of rectangular sheet, $x$ cannot ${\text{b}}$.

equal to 18. 

$x = 5$

${V^{\prime \prime }}(5) = 12(10 - 23) = 12( - 13) =  - 156 < 0$

$x = 5$ is the point of maxima.

19. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Ans: A rectangle of length $l$ and breadth $b$ be inscribed in the given circle of radius $a$ . The diagonal passes through the center and is of length $2a\;{\text{cm}}$.

${(2a)^2} = {l^2} + {b^2}$

$\Rightarrow {b^2} = 4{a^2} - {l^2}$

$\Rightarrow b = \sqrt {4{a^2} - {l^2}} $

$A = l\sqrt {4{a^2} - {l^2}} $

$\therefore \dfrac{{dA}}{{dl}} = \sqrt {4{a^2} - {l^2}}  + l\dfrac{1}{{2\sqrt {4{a^2} - {l^2}} }}( - 2l) = \sqrt {4{a^2} - {l^2}}  - \dfrac{1}{{\sqrt {4{a^2} - {l^2}} }}$

$= \dfrac{{4{a^2} - {l^2}}}{{\sqrt {4{a^2} - {l^2}} }}$

$\dfrac{{{d^2}A}}{{d{l^2}}} = \dfrac{{\sqrt {4{a^2} - {l^2}} ( - 4l) - \left( {4{a^2} - 2{l^2}} \right)\dfrac{{( - 2l)}}{{2\sqrt {4{a^2} - {l^2}} }}}}{{\left( {4{a^2} - {l^2}} \right)}}$

$= \dfrac{{\left( {4{a^2} - {l^2}} \right)( - 4l) + 1\left( {4{a^2} - 2{l^2}} \right)}}{{{{\left( {4{a^2} - {l^2}} \right)}^{\dfrac{3}{2}}}}}$

$= \dfrac{{ - 12{a^2}l + 2{l^3}}}{{{{\left( {4{a^2} - {l^2}} \right)}^{\dfrac{3}{2}}}}} = \dfrac{{ - 2l\left( {6{a^2} - {l^2}} \right)}}{{{{\left( {4{a^2} - {l^2}} \right)}^{\dfrac{3}{2}}}}}$

$\dfrac{{dA}}{{dl}} = 0$ gives $4{a^2} = 2{l^2} \Rightarrow l = \sqrt {2a} $

$\Rightarrow b = \sqrt {4{a^2} - 2{a^2}}  = \sqrt {2{a^2}}  = \sqrt {2a} $

when $l = \sqrt {2a} $,

$\dfrac{{{d^2}A}}{{d{l^2}}} = \dfrac{{ - 2(\sqrt {2a} )\left( {6{a^2} - 2{a^2}} \right)}}{{2\sqrt {2{a^3}} }} = \dfrac{{ - 8\sqrt {2{a^3}} }}{{2\sqrt {2{a^3}} }} =  - 4 < 0$

when $l = \sqrt {2a} $, then area of rectangle is maximum. Since $l = b = \sqrt {2a} $, rectangle is a square.

20. Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base. 

Ans: $S = 2\pi {r^2} + 2\pi rh$

$\Rightarrow h = \dfrac{{S - 2\pi {r^2}}}{{2\pi r}}$

$\quad  = \dfrac{S}{{2\pi }}\left( {\dfrac{1}{r}} \right) - r$

$V = \pi {r^2}h = \pi {r^2}\left[ {\dfrac{S}{{2\pi }}\left( {\dfrac{1}{r}} \right) - r} \right] = \dfrac{{Sr}}{2} =  - \pi {r^3}$

$\dfrac{{dV}}{{dr}} = \dfrac{S}{2} - 3\pi {r^2},\dfrac{{{d^2}V}}{{d{r^2}}} =  - 6\pi r$

$\dfrac{{dV}}{{dr}} = 0 \Rightarrow \dfrac{S}{2} = 3\pi {r^2} \Rightarrow {r^2} = \dfrac{S}{{6\pi }}$

${r^2} = \dfrac{S}{{6\pi }},\dfrac{{{d^2}V}}{{d{r^2}}} =  - 6\pi \left( {\sqrt {\dfrac{S}{{6\pi }}} } \right) < 0$

volume is maximum when ${r^2} = \dfrac{S}{{6\pi }}$. when ${r^2} = \dfrac{S}{{6\pi }}$, 

then $h = \dfrac{{6\pi {r^2}}}{{2\pi }}\left( {\dfrac{1}{r}} \right) - r = 3r - r = 2r$.

21. Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimeters,

find the dimensions of the can which has the minimum surface area?

Ans: $V = \pi {r^2}h = 100$

$\therefore h = \dfrac{{100}}{{\pi {r^2}}}$

$S = 2{\pi ^2} + 2\pi rh = 2\pi {r^2} + \dfrac{{200}}{r}$

$\therefore \dfrac{{dS}}{{dr}} = 4\pi r - \dfrac{{200}}{{{r^2}}},\dfrac{{{d^2}S}}{{d{r^2}}} = 4\pi  + \dfrac{{400}}{{{r^3}}}$

$\dfrac{{dS}}{{dr}} = 0 \Rightarrow 4\pi r = \dfrac{{200}}{{{r^2}}}$

$\Rightarrow {r^3} = \dfrac{{200}}{{4\pi }} = \dfrac{{50}}{\pi }$

$\Rightarrow r = {\left( {\dfrac{{50}}{\pi }} \right)^{\dfrac{1}{3}}}$

when $r = {\left( {\dfrac{{50}}{\pi }} \right)^{\dfrac{1}{3}}},\dfrac{{{d^2}S}}{{d{r^2}}} > 0$.

the surface area is the minimum when the radius of the cylinder is ${\left( {\dfrac{{50}}{\pi }} \right)^{\dfrac{1}{3}}}\;{\text{cm}}$. $r = {\left( {\dfrac{{50}}{\pi }} \right)^{\dfrac{1}{3}}},h = 2{\left( {\dfrac{{50}}{\pi }} \right)^{\dfrac{1}{3}}}\;{\text{cm}}$.

22. A Wire of length $28 m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the circle is minimum? 

Ans: Piece of length $l$ be cut from wire to make square. other piece of wire to be made into circle is $(28 - l)m$. side of square $= \dfrac{l}{4}$

$2\pi r = 28 - l \Rightarrow r = \dfrac{1}{{2\pi }}(28 - l).$

$A = \dfrac{{{l^2}}}{{16}} + \pi {\left[ {\dfrac{1}{{2\pi }}(28 - l)} \right]^2}$

$= \dfrac{{{l^2}}}{{16}} + \dfrac{1}{{4\pi }}{(28 - l)^2}$

$\therefore \dfrac{{dA}}{{dl}} = \dfrac{{2l}}{{16}} + \dfrac{2}{{4\pi }}(28 - l)( - 1) = \dfrac{l}{8} - \dfrac{1}{{2\pi }}(28 - l)$

$\dfrac{{{d^2}A}}{{d{l^2}}} = \dfrac{1}{8} + \dfrac{1}{{2\pi }} > 0$

$\dfrac{{dA}}{{dl}} = 0 \Rightarrow \dfrac{l}{8} - \dfrac{l}{{2\pi }}(28 - l) = 0$

$\Rightarrow \dfrac{{\pi l - 4(28 - l)}}{{8\pi }} = 0$

$(\pi  + 4)l - 112 = 0$

$\Rightarrow l = \dfrac{{112}}{{\pi  + 4}}$

when $l = \dfrac{{112}}{{\pi  + 4}},\dfrac{{{d^2}A}}{{d{l^2}}} > 0$.

the area is minimum when $l = \dfrac{{112}}{{\pi  + 4}}$.

23. Prove that the volume of the largest cone that can be inscribed in a sphere of radius $R$ is $\dfrac{8}{{27}}$ of the volume of the sphere. 

Ans: Let $r$ and $h$ be the radius and height of the cone respectively inscribed in a sphere of radius $R$.

$V = \dfrac{1}{3}\pi {r^2}h$

$h = R + AB = R + \sqrt {{R^2} - {r^2}} $

$\therefore V = \dfrac{1}{3}\pi {r^2}\left( {R + \sqrt {{R^2} - {r^2}} } \right)$

$= \dfrac{1}{3}\pi {r^2}R + \dfrac{1}{3}\pi {r^2}\sqrt {{R^2} + {r^2}} $

$\dfrac{{dV}}{{dr}} = \dfrac{2}{3}\pi rR + \dfrac{2}{3}\pi r\sqrt {{R^2} - {r^2}}  + \dfrac{1}{3}\pi {r^2} \cdot \dfrac{{( - 2r)}}{{2\sqrt {{R^2} - {r^2}} }}$

$= \dfrac{2}{3}\pi rR + \dfrac{2}{3}\pi r\sqrt {{R^2} - {r^2}}  - \dfrac{1}{3}\pi \dfrac{{{r^3}}}{{\sqrt {{R^2} - {r^2}} }}$

$= \dfrac{2}{3}\pi rR + \dfrac{{2\pi r\left( {{R^2} - {r^2}} \right) - \pi {r^3}}}{{3\sqrt {{R^2} - {r^2}} }}$

$= \dfrac{2}{3}\pi rR + \dfrac{{2\pi r{R^2} - 3\pi {r^3}}}{{3\sqrt {{R^2} - {r^2}} }}.$

$\dfrac{{{d^2}V}}{{d{r^2}}} = \dfrac{{2\pi R}}{3} + \dfrac{{3\sqrt {{R^2} - {r^2}} \left( {2\pi {R^2} - 9\pi {r^2}} \right) - \left( {2\pi r{R^2} - 3\pi {r^3}} \right) \cdot \dfrac{{( - 2r)}}{{6\sqrt {{R^2} - {r^2}} }}}}{{9\left( {{R^2} - {r^2}} \right)}}$

$= \dfrac{2}{3}\pi rR + \dfrac{{9\left( {{R^2} - {r^2}} \right)\left( {2\pi {R^2} - 9\pi {r^2}} \right) + 2\pi {r^2}{R^2} + 3\pi {r^4}}}{{27{{\left( {{R^2} - {r^2}} \right)}^{\dfrac{3}{2}}}}}$

$\dfrac{{dV}}{{dr}} = 0 \Rightarrow \pi \dfrac{2}{3}rR = \dfrac{{3\pi {r^3} - 2\pi {R^2}}}{{3\sqrt {{R^2} - {r^2}} }}$

$\Rightarrow 2R = \dfrac{{3\pi {r^3} - 2\pi {R^2}}}{{\sqrt {{R^2} - {r^2}} }} = 2R\sqrt {{R^2} - {r^2}}  = 3{r^2} - 2{R^2}$

$\Rightarrow 4{R^2}\left( {{R^2} - {r^2}} \right) = {\left( {3{r^2} - 2{R^2}} \right)^2}$

$\Rightarrow 4{R^4} - 4{R^2}{r^2} = 9{r^4} + 4{R^4} - 12{r^2}{R^2}$

$\Rightarrow 9{r^4} = 8{R^2}{r^2}$

$\Rightarrow {r^2} = \dfrac{8}{9}{R^2}$

${r^2} = \dfrac{8}{9}{R^2},\dfrac{{{d^2}V}}{{d{r^2}}} < 0$

volume of the cone is the maximum when ${r^2} = \dfrac{8}{9}{R^2}$.

${r^2} = \dfrac{8}{9}{R^2},h = R + \sqrt {{R^2} - \dfrac{8}{9}{R^2}}  = R + \sqrt {\dfrac{1}{9}{R^2}}  = R + \dfrac{R}{3} = \dfrac{4}{3}R$.

$= \dfrac{1}{3}\pi \left( {\dfrac{8}{9}{R^2}} \right)\left( {\dfrac{4}{3}R} \right)$

$= \dfrac{8}{{27}}\left( {\dfrac{4}{3}\pi {R^3}} \right)$

$= \dfrac{8}{{27}}{\text{x}}($ volume of the sphere)

24. Show that he right circular cone of least curved surface and given volume has an altitude equal to $\sqrt 2 $ time the radius of the base.

Ans: $V = \dfrac{1}{{3\pi }}\pi {r^2}h \Rightarrow h = \dfrac{{3V}}{{{r^2}}}$

$S = \pi rl$

$= \pi r\sqrt {{r^2} + {h^2}} $

$= \pi r\sqrt {{r^2} + \dfrac{{9{V^2}}}{{{\pi ^2}{r^4}}}} \pi \dfrac{{r\sqrt {{9^2}{r^6} + {V^2}} }}{{\pi {r^2}}}$

$= \dfrac{1}{r}\sqrt {{\pi ^2}{r^6} + 9{V^2}} $

$\therefore \dfrac{{dS}}{{dr}} = \dfrac{{r \cdot \dfrac{{6{\pi ^2}{r^5}}}{{2\sqrt {{\pi ^2}{r^6} + 9{V^2}} }} - \sqrt {{\pi ^2}{r^6} + 9{V^2}} }}{{{r^2}}}$

$= \dfrac{{3{\pi ^2}{r^6} - {\pi ^2}{r^6} - 9{V^2}}}{{{r^2}\sqrt {{\pi ^2}{r^6} + 9{V^2}} }}$

$= \dfrac{{2{\pi ^2}{r^6} - 9{V^2}}}{{{r^2}\sqrt {{\pi ^2}{r^6} + 9{V^2}} }}$

$= \dfrac{{2{\pi ^2}{r^6} - 9{V^2}}}{{{r^2}\sqrt {{\pi ^2}{r^6} + 9{V^2}} }}$

$\dfrac{{dS}}{{dr}} = 0 \Rightarrow 2{\pi ^2}{r^6} = 9{V^2} \Rightarrow {r^6} = \dfrac{{9{V^2}}}{{2{\pi ^2}}}$

${r^6} = \dfrac{{9{V^2}}}{{2{\pi ^2}}},\dfrac{{{d^2}S}}{{d{r^2}}} > 0$

surface area of the cone is least when ${r^6} = \dfrac{{9{V^2}}}{{2{\pi ^2}}}$

${r^6} = \dfrac{{9{V^2}}}{{2{\pi ^2}}},h = \dfrac{{3V}}{{\pi {r^2}}} = \dfrac{{3V}}{{\pi {r^2}}}{\left( {\dfrac{{2{\pi ^2}{r^6}}}{9}} \right)^{\dfrac{1}{2}}} = \dfrac{3}{{\pi {r^2}}} \cdot \dfrac{{\sqrt {2\pi {r^3}} }}{3} = \sqrt 2 r.$

25. Show that the semi-vertical angle of the cone of the maximum volume and of given slant heigh is ${\tan ^{ - 1}}\sqrt 2 $.

Ans: Let $\theta $ be semi-vertical angle of cone. $\theta  \in \left[ {0,\dfrac{\pi }{2}} \right]$

$r = l\sin \theta $ and $h = l\cos \theta $

$V = \dfrac{1}{3}\pi {r^2}h$

$= \dfrac{1}{3}\pi \left( {{l^2}{{\sin }^2}\theta } \right)(l\cos \theta )$

$= \dfrac{1}{3}\pi {l^3}{\sin ^2}\theta \cos \theta $

$\therefore \dfrac{{dV}}{{d\theta }} = \dfrac{{{l^3}\pi }}{3}\left[ {{{\sin }^2}\theta ( - \sin \theta ) + \cos \theta (2\sin \theta \cos \theta )} \right]$

\[ = \dfrac{{{l^3}\pi }}{3}\left[ { - {{\sin }^3} + 2\sin \theta {{\cos }^2}\theta } \right]\]

\[\dfrac{{{d^2}V}}{{d{\theta ^2}}} = \dfrac{{{l^3}\pi }}{3}\left[ { - 3{{\sin }^2}\theta \cos \theta  + 2{{\cos }^3}\theta  - 4{{\sin }^2}\theta \cos \theta } \right]\]

\[ = \dfrac{{{l^3}\pi }}{3}\left[ {2{{\cos }^2}\theta  - 7{{\sin }^2}\theta \cos \theta } \right]\]

\[\dfrac{{dV}}{{d\theta }} = 0\]

$\Rightarrow {\sin ^3}\theta  = 2\sin \theta {\cos ^2}\theta $

$\Rightarrow {\tan ^2}\theta  = 2$

$\Rightarrow \tan \theta  = \sqrt 2 $

$\Rightarrow \theta  = {\tan ^{ - 1}}\sqrt 2 $

when $\theta  = {\tan ^{ - 1}}\sqrt 2 $, then ${\tan ^2}\theta  = 2$ or ${\sin ^2}\theta  = 2{\cos ^2}\theta $.

$\dfrac{{{d^2}V}}{{d{\theta ^2}}} = \dfrac{{{l^3}\pi }}{3}\left[ {2{{\cos }^3}\theta  - 14{{\cos }^3}\theta } \right] =  - 4\pi {l^3}{\cos ^3}\theta  < 0$ for $\theta  \in \left[ {0,\dfrac{\pi }{2}} \right]$

volume is the maximum when $\theta  = {\tan ^{ - 1}}\sqrt 2 $. 

26. The point on the curve ${x^2} = 2y$ which is nearest to the point $(0,5)$ is

${{\text{ (A) }}(2\sqrt 2 ,4)}{{\text{ (B) }}(2\sqrt 2 ,0)}{{\text{ (C) }}(0,0)}{{\text{ (D) }}(2,2)}$

Ans: position of point is $\left( {x,\dfrac{{{x^2}}}{2}} \right)$. 

distance $d(x)$ between points $\left( {x,\dfrac{{{x^2}}}{2}} \right)$ and $(0,5)$ is $d(x) = \sqrt {{{(x - 0)}^2} + {{\left( {\dfrac{{{x^2}}}{2} - 5} \right)}^2}}  = \sqrt {{x^2} + \dfrac{{{x^4}}}{4} + 25 - 5{x^2}}  = \sqrt {\dfrac{{{x^4}}}{4} - 4{x^2} + 25} $

$\therefore {d^\prime }(x) = \dfrac{{\left( {{x^3} - 8x} \right)}}{{2\sqrt {\dfrac{{{x^4}}}{4} - 4{x^2} + 25} }} = \dfrac{{\left( {{x^3} - 8x} \right)}}{{\sqrt {{x^4} - 16{x^2} + 100} }}$

${d^\prime }(x) = 0 \Rightarrow {x^3} - 8x = 0$

$\Rightarrow x\left( {{x^2} - 8} \right) = 0$

$\Rightarrow x = 0, \pm 2\sqrt 2 $

\[ = \dfrac{{\left( {{x^4} - 16{x^2} + 100} \right)\left( {3{x^2} - 8} \right) - 2\left( {{x^3} - 8x} \right)\left( {{x^3} - 8x} \right)}}{{{{\left( {{x^4} - 16{x^2} + 100} \right)}^{\dfrac{3}{2}}}}}\]

\[ = \dfrac{{\left( {{x^4} - 16{x^2} + 100} \right)\left( {3{x^2} - 8} \right) - 2{{\left( {{x^3} - 8x} \right)}^2}}}{{{{\left( {{x^4} - 16{x^2} + 100} \right)}^{\dfrac{3}{2}}}}}\]

$x = 0$, then ${d^{\prime \prime }}(x) = \dfrac{{36( - 8)}}{{{6^3}}} < 0.$

$x =  \pm 2\sqrt 2 ,{d^{\prime \prime }}(x) > 0$

$d(x)$ is the minimum at $x =  \pm 2\sqrt 2 $. $x =  \pm 2\sqrt 2 ,y = \dfrac{{{{(2\sqrt 2 )}^2}}}{2} = 4.$

The correct answer is A. 

27. For all real values of $x$, the minimum value of $\dfrac{{1 - x + {x^2}}}{{1 + x + {x^2}}}$ is

1. 0 (B) $1({\mathbf{C}})3$ (D) $\dfrac{1}{3}$

Ans: $f(x) = \dfrac{{1 - x + {x^2}}}{{1 + x + {x^2}}}$

$\therefore {f^\prime }(x) = \dfrac{{\left( {1 - x + {x^2}} \right)( - 1 + 2x) - \left( {1 - x + {x^2}} \right)(1 + 2x)}}{{{{\left( {1 + x + {x^2}} \right)}^2}}}$

$= \dfrac{{ - 1 + 2x - x + 2{x^2} - {x^2} + 2{x^2} - 1 - 2x + x + 2{x^2} - {x^2} - 2{x^3}}}{{{{\left( {1 + x + {x^2}} \right)}^2}}}$

$= \dfrac{{2{x^2} - 2}}{{{{\left( {1 + x + {x^2}} \right)}^2}}} = \dfrac{{2\left( {{x^2} - 1} \right)}}{{{{\left( {1 + x + {x^2}} \right)}^2}}}$

$\therefore {f^\prime }(x) = 0 \Rightarrow {x^2} = 1 \Rightarrow x =  \pm 1$

${f^{\prime \prime }}(x) = \dfrac{{2\left[ {\left( {1 + x + {x^2}} \right)(2x) - \left( {{x^2} - 1} \right)(2)\left( {1 + x + {x^2}} \right)(1 + 2x)} \right]}}{{{{\left( {1 + x + {x^2}} \right)}^4}}}$

$= \dfrac{{4\left( {1 + x + {x^2}} \right)\left[ {\left( {1 + x + {x^2}} \right)x - \left( {{x^2} - 1} \right)(1 + 2x)} \right]}}{{{{\left( {1 + x + {x^2}} \right)}^4}}}$

$= 4\dfrac{{\left[ {x + {x^2} + {x^3} - {x^2} - 2{x^3} + 1 + 2x} \right]}}{{{{\left( {1 + x + {x^2}} \right)}^3}}}$

$= \dfrac{{4\left( {1 + 3x - {x^3}} \right)}}{{{{\left( {1 + x + {x^2}} \right)}^3}}}$

${f^{\prime \prime }}(1) = \dfrac{{4(1 + 3 - 1)}}{{{{(1 + 1 + 1)}^3}}} = \dfrac{{4(3)}}{{{{(3)}^3}}} = \dfrac{4}{9} > 0$

${f^{\prime \prime }}( - 1) = \dfrac{{4(1 - 3 + 1)}}{{{{(1 + 1 + 1)}^3}}} = 4( - 1) = 4 < 0$

$f$ is the minimum at $x = 1$ and the minimum value is given by

$f(1) = \dfrac{{1 - 1 + 1}}{{1 + 1 + 1}} = \dfrac{1}{3}$.

The correct answer is ${\mathbf{D}}$.

28. The maximum value of ${[x(x + 1) + 1]^{\dfrac{1}{3}}},0 \leqslant x \leqslant 1$ is

1. ${\left( {\dfrac{1}{3}} \right)^{\dfrac{1}{3}}}$ 

2. $\dfrac{1}{2}$

$({\mathbf{C}})1{\kern 1pt} \,$

$({\mathbf{D}})0$

Ans: $f(x) = {[x(x + 1) + 1]^{\dfrac{1}{3}}}$

$\therefore {f^\prime }(x) = \dfrac{{2x - 1}}{{3{{[x(x + 1) + 1]}^{\dfrac{2}{3}}}}}$

${f^\prime }(x) = 0 \Rightarrow x = \dfrac{1}{2}$

$f(0) = {[0(0 - 1) + 1]^{\dfrac{1}{3}}} = 1$

$f(1) = {[1(1 - 1) + 1]^{\dfrac{1}{3}}} = 1$

$f\left( {\dfrac{1}{2}} \right) = {\left[ {\dfrac{1}{2}\left( {\dfrac{{ - 1}}{2}} \right) + 1} \right]^{\dfrac{1}{3}}} = {\left( {\dfrac{3}{4}} \right)^{\dfrac{1}{3}}}$

Maximum value of $f$ in \[[0,1]\]is 1. 

The correct answer is ${\mathbf{C}}$.

Miscellaneous Solutions:

1. Using differentials, find the approximate value of each of the following.

1. ${\left( {\dfrac{{17}}{{81}}} \right)^{\dfrac{1}{4}}}$

2. ${(33)^{\dfrac{1}{5}}}$

Ans:

1. $y = {x^{\dfrac{1}{4}}}$

$x = \dfrac{{16}}{{81}}$

$\Delta x = \dfrac{1}{{81}}$.

$\Delta y = {(x + \Delta x)^{\dfrac{1}{4}}} - {x^{\dfrac{1}{4}}}$

$= {\left( {\dfrac{{17}}{{81}}} \right)^{\dfrac{1}{4}}} - {\left( {\dfrac{{16}}{{81}}} \right)^{\dfrac{1}{4}}}$

$= {\left( {\dfrac{{17}}{{81}}} \right)^{\dfrac{1}{4}}} - \dfrac{2}{3}$

$= {\left( {\dfrac{{17}}{{81}}} \right)^{\dfrac{1}{4}}} = \dfrac{2}{3} + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x = \dfrac{1}{{4{{(x)}^{\dfrac{3}{4}}}}}(\Delta x)$

$= \dfrac{1}{{4{{\left( {\dfrac{{16}}{{81}}} \right)}^{\dfrac{3}{4}}}}}\left( {\dfrac{1}{{81}}} \right) = \dfrac{{27}}{{4 \times 8}} \times \dfrac{1}{{81}} = \dfrac{1}{{32 \times 3}} = \dfrac{1}{{96}} = 0.010$

approximate value of ${\left( {\dfrac{{17}}{{81}}} \right)^{\dfrac{1}{4}}}$ is $\dfrac{2}{3} + 0.010 = 0.667 + 0.010$

$= 0.677$.

2. $y = {x^{\dfrac{1}{5}}}$

$x = 32$

$\Delta x = 1.$

$\Delta y = {(x + \Delta x)^{\dfrac{1}{5}}} - {x^{\dfrac{1}{5}}} = {(33)^{\dfrac{1}{5}}} - {(32)^{\dfrac{1}{5}}} = {(33)^{ - \dfrac{1}{5}}} - \dfrac{1}{2}$

$\therefore {(33)^{\dfrac{1}{5}}} = \dfrac{1}{2} + \Delta y$

$dy = \left( {\dfrac{{dy}}{{dx}}} \right) = (\Delta x) = \dfrac{{ - 1}}{{5{{(x)}^{\dfrac{6}{5}}}}}(\Delta x)$

$= \dfrac{1}{{5{{(2)}^6}}}(1) = \dfrac{1}{{320}} =  - 0.003$

approximate value of ${(33)^{\dfrac{1}{5}}}$ is $\dfrac{1}{2} + ( - 0.003) = 0.5 - 0.003 = 0.497$.

2. Show that the function given by $f(x) = \dfrac{{\log x}}{x}$ has maximum at $x = e$.

Ans: $f(x) = \dfrac{{\log x}}{x}$

${f^\prime }(x) = \dfrac{{x\left( {\dfrac{1}{x}} \right) - \log x}}{{{x^2}}} = \dfrac{{1 - \log x}}{{{x^2}}}$

${f^\prime }(x) = 0$

$\Rightarrow 1 - \log x = 0$

$\Rightarrow \log x = 1$

$\Rightarrow \log x = \log e$

$\Rightarrow x = e$

${f^{\prime \prime }}(x) = \dfrac{{{x^2}\left( { - \dfrac{1}{x}} \right) - (1 - \log x)(2x)}}{{{x^4}}}$

$= \dfrac{{ - x - 2x(1 - \log x)}}{{{x^4}}}$

$= \dfrac{{ - 3 + 2\log x}}{{{x^3}}}$

${f^{\prime \prime }}(e) =  = \dfrac{{ - 3 + 2\log e}}{{{e^3}}} = \dfrac{{ - 3 + 2}}{{{e^3}}} = \dfrac{{ - 1}}{{{e^3}}} < 0$

$f$ is the maximum at $x = e$

3. The two equal sides of an isosceles triangle with fixed base $b$ are decreasing at the rate of $3\;{\text{cm}}$ per second. How fast is the area decreasing when the two equal sides are equal to the base? 

Ans: Let $\vartriangle ABC$ be isosceles where \[BC\]is the base of fixed length $b$. 

Let the length of the two equal sides of $\vartriangle ABC$ be $a$. 

Draw $AD \bot BC$.

$AD = \sqrt {{a^2} - \dfrac{{{b^2}}}{4}} $

Area of triangle $= \dfrac{1}{2}b\sqrt {{a^2} - \dfrac{{{b^2}}}{4}} $

$\dfrac{{dA}}{{dt}} = \dfrac{1}{2}b \cdot \dfrac{{2a}}{{2\sqrt {{a^2} - \dfrac{{{b^2}}}{4}} }}\dfrac{{da}}{{dt}} = \dfrac{{ab}}{{\sqrt {4{a^2} - {b^2}} }}\dfrac{{da}}{{dt}}$

$\dfrac{{da}}{{dt}} =  - 3\;{\text{cm}}/{\text{s}}$

$\therefore \dfrac{{da}}{{dt}} = \dfrac{{ - 3ab}}{{\sqrt {4{a^2} - {b^2}} }}$

when $a = b$

$\dfrac{{dA}}{{dt}} = \dfrac{{ - 3{b^2}}}{{\sqrt {4{b^2} - {b^2}} }} = \dfrac{{ - 3{b^2}}}{{\sqrt {3{b^2}} }} =  - \sqrt {3b}$

4.Find the equation of the normal to curve ${y^2} = 4x$ at the point $(1,2)$. 

Ans: $2y\dfrac{{dy}}{{dx}} = 4$

$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{4}{{2y}} = \dfrac{2}{y}$

${\left. {\therefore \dfrac{{dy}}{{dx}}} \right]_{(1,2)}} = \dfrac{2}{2} = 1$

slope of the normal at $(1,2)$ is $\dfrac{{ - 1}}{{{{\left. {\dfrac{{dy}}{{dx}}} \right]}_{(1,2)}}}} = \dfrac{{ - 1}}{1} =  - 1$. 

Equation of normal at $(1,2)$ is $y - 2 =  - 1(x - 1)$. $y - 2 =  - x + 1$

$x + y - 3 = 0$

5. Show that the normal at any point $\theta $ to the curve $x = a\cos \theta  + a\theta \sin \theta ,$$y = a\sin \theta  - a\theta \cos \theta $ is at a constant distance from the origin.

Ans: $x = a\cos \theta  + a\theta \sin \theta $.

$\therefore \dfrac{{dx}}{{d\theta }} =  - a\sin \theta  + a\sin \theta  + a\theta \cos \theta  = a\theta \cos \theta $

$y = a\sin \theta  - a\theta \cos \theta $

$\therefore \dfrac{{dy}}{{d\theta }} = a\cos \theta  - a\cos \theta  + a\theta \sin \theta  = a\theta \sin \theta $

$\therefore \dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{d\theta }} \cdot \dfrac{{d\theta }}{{dx}} = \dfrac{{a\theta \sin \theta }}{{a\theta \cos \theta }} = \tan \theta $

Slope of normal at any point $\theta $ is $\dfrac{1}{{\tan \theta }}$. equation of normal at a given point $(x,y)$ is given by,

$y - a\sin \theta  + a\theta \cos \theta  = \dfrac{{ - 1}}{{\tan \theta }}(x - a\cos \theta  - a\theta \sin \theta )$

$\Rightarrow y\sin \theta  - a{\sin ^2}\theta  + a\theta \sin \theta \cos \theta  =  - x\cos \theta  + a{\cos ^2}\theta  + a\theta \sin \theta \cos \theta $

$\Rightarrow x\cos \theta  + y\sin \theta  - a\left( {{{\sin }^2}\theta  + {{\cos }^2}\theta } \right) = 0$

$\Rightarrow x\cos \theta  + y\sin \theta  - a = 0$

Perpendicular distance of normal from origin is $\dfrac{{| - a|}}{{\sqrt {{{\cos }^2}} \theta  + {{\sin }^2}\theta }} = \dfrac{{| - a|}}{{\sqrt 1 }} = | - a|$, which is independent of $\theta $.

Perpendicular distance of normal from the origin is constant.

6. Find the intervals in which the function $f$ given by $f(x) = \dfrac{{4\sin x - 2x - x\cos x}}{{2 + \cos x}}$ Is 

(i) increasing (ii) decreasing

Ans: $f(x) = \dfrac{{4\sin x - 2x - x\cos x}}{{2 + \cos x}}$

$\therefore {f^\prime }(x) = \dfrac{{(2 + \cos x)(4\cos x - 2 - \cos x + x\sin x) - (4\sin x - 2x - x\cos x)( - \sin x)}}{{{{(2 + \cos x)}^2}}}$

$= \dfrac{{(2 + \cos x)(3\cos x - 2 + x\sin x) + \sin x(4\sin x - 2x - x\cos x)}}{{{{(2 + \cos x)}^2}}}$

$= \dfrac{{6\cos x - 4 + 2x\sin x + 3{{\cos }^2}x - 2\cos x + x\sin x\cos x + 4{{\sin }^2}x - 2{{\sin }^2}x - 2x\sin x - x\sin x\cos x}}{{{{(2 + \cos x)}^2}}}$

$= \dfrac{{4\cos x - 4 + 3{{\cos }^2}x + 4{{\sin }^2}x}}{{{{(2 + \cos x)}^2}}}$

$= \dfrac{{4\cos x - {{\cos }^2}x}}{{{{(2 + \cos x)}^2}}} = \dfrac{{\cos x(4 - \cos x)}}{{{{(2 + \cos x)}^2}}}$

${f^\prime }(x) = 0$

$\Rightarrow \cos x = 0\cos x = 4$

$\cos x \ne 4$

$\cos x = 0$

$\Rightarrow x = \dfrac{\pi }{2},\dfrac{{3\pi }}{2}$

$\operatorname{In} \left( {0,\dfrac{\pi }{2}} \right)$ and $\left( {\dfrac{{3\pi }}{2},2\pi } \right),{f^\prime }(x) > 0$

$f(x)$ is increasing for $0 < x < \dfrac{x}{2}$ and $\dfrac{{3\pi }}{2} < x < 2\pi $.

$\operatorname{In} \left( {\dfrac{\pi }{2},\dfrac{{3\pi }}{2}} \right),{f^\prime }(x) < 0$

$f(x)$ is decreasing for $\dfrac{\pi }{2} < x < \dfrac{{3\pi }}{2}$.

7. Find the intervals in which the function $f$ given by $f(x) = {x^3} + \dfrac{1}{{{x^3}}},x \ne 0$ is

(i) increasing 

(ii) decreasing

Ans: $f(x) = {x^3} + \dfrac{1}{{{x^3}}}$

$\therefore {f^\prime }(x) = 3{x^2} - \dfrac{3}{{{x^4}}} = \dfrac{{3{x^6} - 3}}{{{x^4}}}$

${f^\prime }(x) = 0 \Rightarrow 3{x^6} - 3 = 0 \Rightarrow {x^6} = x =  \pm 1$

In $( - \infty ,1)$ and $(1,\infty )$ i.e.., when $x <  - 1$ and $x > 1,{f^\prime }(x) > 0.$ when $x <  - 1$ and $x > 1,f$ is increasing. In $( - 1,1)$ i.e., when $ - 1 < x < 1,{f^\prime }(x) < 0$.

Thus, when $ - 1 < x < 1,f$ is decreasing.

8. Find the maximum area of an isosceles triangle inscribed in the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ with its vertex at one end of the major axis.

Ans:

Ellipse $\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1$

Let $A B C$, be the triangle inscribed in the ellipse where vertex $C$ is at $(a,0)$. Since the ellipse is symmetrical with $x$-axis and $y - axis$ ${y_1} =  \pm \dfrac{b}{a}\sqrt {{a^2} - x_1^2} .$

Coordinates of $A$ are $\left( { - {x_{1,}}\dfrac{b}{a}\sqrt {{a^2} - {x^2}1} } \right)$ and coordinates of $B$ are $\left( {{x_1}, - \dfrac{b}{a}\sqrt {{a^2} - x_1^2} .} \right)$ As the point $\left( { - {x_1},{y_1}} \right)$ lies on the ellipse, the area of triangle $A B C$ is $A = \dfrac{1}{2}\mid a\left( {\dfrac{{2b}}{a}\sqrt {{a^2} - x_1^2} } \right) + \left( { - {x_1}} \right)\left( { - \dfrac{b}{a}\sqrt {{a^2} - x_1^2} } \right) + \left( { - {x_1}} \right)\left( { - \dfrac{b}{a}\sqrt {{a^2} - {x^2}1} } \right)$

$\Rightarrow A = ba\sqrt {{a^2} - {x^2}1}  + {x_1}\dfrac{b}{a}\sqrt {{a^2} - x_1^2} $

$\therefore \dfrac{{dA}}{{d{x_1}}} = \dfrac{{ - 2xb}}{{2\sqrt {{a^2} - x_1^2} }} + \dfrac{b}{a}\sqrt {{a^2} - x_1^2}  - \dfrac{{2bx_1^2}}{{{a^2}\sqrt {{a^2} - x_1^2} }}$

$= \dfrac{b}{{2\sqrt {{a^2} - x_1^2} }}\left[ { - {x_1}a + \left( {{a^2} - x_1^2} \right) - x_1^2} \right]$

$= \dfrac{{b\left( { - 2x_1^2 - x_1^2 + {a^2}} \right)}}{{a\sqrt {{a^2} - x_1^2} }}$

$\dfrac{{dA}}{{d{x_1}}} = 0$

$\Rightarrow  - 2x_1^2 - {x_1}a + {a^2} = 0$

$\Rightarrow {x_1} = \dfrac{{a \pm \sqrt {{a^2} - 4( - 2)\left( {{a^2}} \right)} }}{{2( - 2)}}$

$= \dfrac{{a \pm \sqrt {9{a^2}} }}{{ - 4}}$

$= \dfrac{{a \pm 3a}}{{ - 4}}$

$\Rightarrow {x_1} =  - a,\dfrac{a}{2}$

${x_1}$ cannot be equal to $a$. $\therefore {x_1} = \dfrac{a}{2} \Rightarrow {y_1} = \dfrac{b}{a}\sqrt {{a^2} - \dfrac{{{a^2}}}{4}}  = \dfrac{{ba}}{{2a}}\sqrt 3  = \dfrac{{\sqrt {3b} }}{2}$

Now,$\dfrac{{{d^2}A}}{{d{x^2}1}} = \dfrac{b}{a}\left\{ {\dfrac{{\sqrt {{a^2} - x_1^2} \left( { - 4{x_1} - a} \right) - \left( { - 2x_1^2 - {x_1}a + {a^2}} \right)\dfrac{{\left( { - 2{x_1}} \right)}}{{2\sqrt {{a^2} - x_1^2} }}}}{{{a^2} - x_1^2}}} \right\}$

$= \dfrac{b}{a}\left\{ {\dfrac{{\left( {{a^2} - x_1^2} \right)\left( { - 4{x_1} - a} \right) + {x_1}\left( { - 2x_1^2 - {x_1}a + {a^2}} \right)}}{{{{\left( {{a^2} - x_1^2} \right)}^{\dfrac{2}{3}}}}}} \right\}$

$= \dfrac{b}{a}\left\{ {\dfrac{{2{x^3} - 3{a^2}x - {a^3}}}{{{{\left( {{a^2} - x_1^2} \right)}^{\dfrac{3}{2}}}}}} \right\}$

when ${x_1} = \dfrac{a}{2}$,

$\dfrac{{{d^2}A}}{{dx_1^2}} = \dfrac{b}{a}\left\{ {\dfrac{{2\dfrac{{{a^3}}}{8} - 3\dfrac{{{a^3}}}{2} - {a^3}}}{{{{\left( {\dfrac{{3{a^2}}}{4}} \right)}^{\dfrac{3}{2}}}}}} \right\} = \dfrac{b}{a}\left\{ {\dfrac{{\dfrac{{{a^3}}}{4} - \dfrac{3}{2}{a^3} - {a^3}}}{{{{\left( {\dfrac{{3{a^2}}}{4}} \right)}^{\dfrac{3}{2}}}}}} \right\}$

$= \dfrac{b}{a}\left\{ {\dfrac{{\dfrac{9}{4}{a^3}}}{{{{\left( {\dfrac{{3{a^2}}}{4}} \right)}^{\dfrac{3}{2}}}}}} \right\} < 0$

Area is the maximum when ${x_1} = \dfrac{a}{2}$. Maximum area of the triangle is $A = b\sqrt {{a^2} - \dfrac{{{a^2}}}{4}}  + \left( {\dfrac{a}{2}} \right)\dfrac{b}{a}\sqrt {{a^2} - \dfrac{{{a^2}}}{4}} $

$= ab\dfrac{{\sqrt 3 }}{2} + \left( {\dfrac{a}{2}} \right)\dfrac{b}{a} \times \dfrac{{a\sqrt 3 }}{2}$

$= \dfrac{{ab\sqrt 3 }}{2} + \dfrac{{ab\sqrt 3 }}{4} = \dfrac{{3\sqrt 3 }}{4}ab$

9. A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is \[2m\]and volume is $8{m^3}$. If the building of tank costs Rs 70 per sq meters for the base and Rs 45 per sq meters for sides. What is the cost of the least expensive tank?

Ans: Let \[l,b\]and $h$ represent the length, breadth, and height of the tank respectively.

height $(h) = 2m$

Volume of the tank $= 8\;{{\text{m}}^3}$ Volume of the tank $= l \times b \times {\text{h}}$ $8 = l \times b \times 2$

$\Rightarrow lb = 4 \Rightarrow b = \dfrac{4}{l}$

Area of the base $= lb = 4$ 

Area of 4 walls $(A) = 2h(l + b)$

$\therefore A = 4\left( {l + \dfrac{4}{l}} \right)$

$\Rightarrow \dfrac{{dA}}{{dl}} = 4\left( {1 - \dfrac{4}{{{l^2}}}} \right)$

$Now,\dfrac{{dA}}{{dl}} = 0$

$\Rightarrow l - \dfrac{4}{{{l^2}}} = 0$

$\Rightarrow {l^2} = 4$

$\Rightarrow l =  \pm 2$

Therefore, we have $l = 4$. 

$\therefore b = \dfrac{4}{l} = \dfrac{4}{2} = 2$

$\dfrac{{{d^2}A}}{{d{l^2}}} = \dfrac{{32}}{{{l^3}}}$

$l = 2,\dfrac{{{d^2}A}}{{d{l^2}}} = \dfrac{{32}}{8} = 4 > 0.$

Area is the minimum when $l = 2$. 

We have $l = b = h = 2$.

Cost of building base \[ = \] Rs $70{\text{x}}(lb) = $ Rs$70(4) = $ Rs 280

Cost of building walls $= Rs2h(l + h) \times 45 = Rs90(2)(2 + 2) = $ 

Rs $8(90) = $ Rs 720

Required total cost $= Rs(280 + 720) = $ Rs 1000

10. The sum of the perimeter of a circle and square is $k$, where $k$ is some constant. Prove that the sum of their area is least when the side of square is double the radius of the circle. 

Ans: $2\pi r + 4a = k$ (where $k$ is constant) $\Rightarrow a = \dfrac{{k - 2\pi r}}{4}$

sum of the areas of the circle and the square $(A)$ is given by,

$A = \pi {r^2} + {a^2} = \pi {r^2} + \dfrac{{{{(k = 2\pi r)}^2}}}{{16}}$

$\therefore \dfrac{{dA}}{{dr}} = 2\pi r + \dfrac{{2(k - 2\pi r)(2\pi )}}{{16}} = 2\pi r$

$=  - \dfrac{{\pi (k - 2\pi r)}}{4}$

Now, $\dfrac{{dA}}{{dr}} = 0$

$\Rightarrow 2\pi r = \dfrac{{\pi (k - 2\pi r)}}{4}$

$8r = k - 2\pi r$

$\Rightarrow (8 + 2\pi )r = k$

$\Rightarrow r = \dfrac{k}{{8 + 2\pi }} = \dfrac{k}{{2(4 + \pi )}}$

Now, $\dfrac{{{d^2}A}}{{d{r^2}}} = 2\pi  + \dfrac{{{\pi ^2}}}{2} > 0$

$\therefore $ where $r = \dfrac{k}{{2(4 + \pi )}},\dfrac{{{d^2}A}}{{d{r^2}}} > 0.$

area is least when $r = \dfrac{k}{{2(4 + \pi )}}$ where $r = \dfrac{k}{{2(4 + \pi )}},$

$a = \dfrac{{k - 2\pi \left[ {\dfrac{k}{{2(4 + \pi )}}} \right]}}{4} = \dfrac{{8k + 2\pi k - 2\pi k}}{{2(4 + \pi ) \times 4}} = \dfrac{k}{{4 + \pi }} = 2r$

11. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10\;{\text{m}}$. Find the dimensions of the window to admit maximum light through the whole opening. 

Ans: $x$ and $y$ be length and breadth of the rectangular window.

Radius of semicircular opening $= \dfrac{x}{2}$

$\therefore x + 2y + \dfrac{{\pi {\text{x}}}}{2} = 10$

$\Rightarrow x\left( {1 + \dfrac{\pi }{2}} \right) + 2y = 10$

$\Rightarrow 2y = 10 - x\left( {1 + \dfrac{\pi }{2}} \right)$

$\Rightarrow y = 5 - x\left( {\dfrac{1}{2} + \dfrac{\pi }{4}} \right)$

$A = xy + \dfrac{\pi }{2}{\left( {\dfrac{x}{2}} \right)^2}$

$= x\left[ {5 - x\left( {\dfrac{1}{2} + \dfrac{\pi }{4}} \right)} \right] + \dfrac{\pi }{8}{\text{x}}{x^2}$

$= 5x - {x^2}\left( {\dfrac{1}{2} + \dfrac{\pi }{4}} \right) + \dfrac{\pi }{8}{\text{x}}{x^2}$

$\therefore \dfrac{{dA}}{{dx}} = 5 - 2x\left( {\dfrac{1}{2} + \dfrac{\pi }{4}} \right) + \dfrac{\pi }{4}x$

$\dfrac{{{d^2}A}}{{d{x^2}}} =  - \left( {1 - \dfrac{\pi }{2}} \right) + \dfrac{\pi }{4} =  - 1 - \dfrac{\pi }{4}$

$\dfrac{{dA}}{{dx}} = 0$

$\Rightarrow 5 - x\left( {1 + \dfrac{\pi }{2}} \right) + \dfrac{\pi }{4}x = 0$

$\Rightarrow 5 - x - \dfrac{\pi }{4}x = 0$

$\Rightarrow x\left( {1 + \dfrac{\pi }{4}} \right) = 5$

$\Rightarrow x = \dfrac{5}{{\left( {1 + \dfrac{\pi }{4}} \right)}} = \dfrac{{20}}{{\pi  + 4}}$

$x = \dfrac{{20}}{{\pi  + 4}},\dfrac{{{d^2}A}}{{d{x^2}}} < 0.$

area is maximum when length $x = \dfrac{{20}}{{\pi  + 4}}m$. 

Now, $y = 5 - \dfrac{{20}}{{\pi  + 4}}\left( {\dfrac{{2 + \pi }}{4}} \right) = 5 - \dfrac{{5(2 + \pi )}}{{\pi  + 4}} = \dfrac{{10}}{{\pi  + 4}}m$

the required dimensions length $= \dfrac{{20}}{{\pi  + 4}}m$ and breadth $= \dfrac{{10}}{{\pi  + 4}}m$.

12. A point of the hypotenuse of a triangle is at distance $a$ and $b$ from the sides of the triangle. Show that the minimum length of the hypotenuse is ${\left( {{a^{\dfrac{2}{3}}} + {b^{\dfrac{2}{3}}}} \right)^{\dfrac{3}{2}}}$ 

Ans: $\vartriangle ABC$ right-angled at $B$. $AB = x$ and $BC = y$.

$P$ be a point on hypotenuse such that ${\text{P}}$ is at a distance of a and ${\text{b}}$ from the sides \[AB\]and \[BC\]respectively.

$\angle c = \theta $

$AC = \sqrt {{x^2} + {y^2}} $

$PC = b\operatorname{cosec} \theta $

$AP = a\sec \theta $

$AC = AP + PC$

$AC = b\operatorname{cosec} \theta  + a\sec \theta  \ldots ..(1)$

$\therefore \dfrac{{d(AC)}}{{d\theta }} =  - b\operatorname{cosec} \theta \cot \theta  + a\sec \theta \tan \theta $

$\therefore \dfrac{{d(AC)}}{{d\theta }} = 0$

$\Rightarrow a\sec \theta \tan \theta  = b\cos \operatorname{ec} \theta \cot \theta $

$\Rightarrow \dfrac{a}{{\cos \theta }} \cdot \dfrac{{\sin \theta }}{{\cos \theta }} = \dfrac{b}{{\sin \theta }}\dfrac{{\cos \theta }}{{\sin \theta }}$

$\Rightarrow a{\sin ^3}\theta  = b{\cos ^3}\theta $

$\Rightarrow {(a)^{\dfrac{1}{3}}}\sin \theta  = {(b)^{\dfrac{1}{3}}}\cos \theta $

$\Rightarrow \tan \theta  = {\left( {\dfrac{b}{a}} \right)^{\dfrac{1}{3}}}$

$\therefore \sin \theta  = \dfrac{{{{(b)}^{\dfrac{1}{3}}}}}{{\sqrt {{a^{\dfrac{2}{3}}} + {b^{\dfrac{2}{3}}}} }}$ and $\cos \theta  = \dfrac{{{{(a)}^{\dfrac{1}{3}}}}}{{\sqrt {{a^{\dfrac{2}{3}}} + {b^{\dfrac{2}{3}}}} }}\quad  \ldots ..(2)$

clearly $\dfrac{{{d^2}(AC)}}{{d{\theta ^2}}} < 0$ when $\tan \theta  = {\left( {\dfrac{b}{a}} \right)^{\dfrac{1}{3}}}$.

the length of the hypotenuse is the maximum when $\tan \theta  = {\left( {\dfrac{b}{a}} \right)^{\dfrac{1}{3}}}$. 

Now, when $\tan \theta  = {\left( {\dfrac{b}{a}} \right)^{\dfrac{1}{3}}}$

$\tan \theta  = {\left( {\dfrac{b}{a}} \right)^{\dfrac{1}{3}}}$

$AC = \dfrac{{b\sqrt {{a^{\dfrac{2}{3}}} + {b^{\dfrac{2}{3}}}} }}{{{b^{\dfrac{1}{3}}}}} + \dfrac{{a\sqrt {{a^{\dfrac{2}{3}}} + {b^{\dfrac{2}{3}}}} }}{{{a^{\dfrac{1}{3}}}}}$

$= a\sqrt {{a^{\dfrac{2}{3}}} + {b^{\dfrac{2}{3}}}} \left( {{b^{\dfrac{2}{3}}} + {a^{\dfrac{2}{3}}}} \right)$

$= {\left( {{a^{\dfrac{2}{3}}} + {b^{\dfrac{2}{3}}}} \right)^{\dfrac{3}{2}}}$

maximum length of the hypotenuse is $= {\left( {{a^{\dfrac{2}{3}}} + {b^{\dfrac{2}{3}}}} \right)^{\dfrac{3}{2}}}$.

13. Find the points at which the function $f$ given by $f(x) = {(x - 2)^4}{(x + 1)^3}$ has

(i) local maxima

(ii) local minima

(iii) point of inflexion

Ans: $f(x) = {(x - 2)^4}{(x + 1)^3}$

$\therefore {f^\prime }(x) = 4{(x - 2)^3}{(x + 1)^3} + 3{(x + 1)^2}{(x - 2)^4}$

$= {(x - 2)^3}{(x + 1)^2}[4(x + 1) + 3(x - 2)]$

$= {(x - 2)^3}{(x + 1)^2}(7x - 2)$

${f^\prime }(x) = 0 \Rightarrow x =  - 1$ and $x = \dfrac{2}{7}$ or $x = 2$

for $x$ close to $\dfrac{2}{7}$ and to left of $\dfrac{2}{7},{f^\prime }(x) > 0$.

for $x$ close to $\dfrac{2}{7}$ and to right of $\dfrac{2}{7},{f^\prime }(x) > 0$. $x = \dfrac{2}{7}$ is point of local minima.

as the value of $x$ varies ${f^\prime }(x)$ does not changes its sign.

$x =  - 1$ is point of inflexion.

14. Find the absolute maximum and minimum values of the function $f$ given by

$f(x) = {\cos ^2}x + \sin x,x \in [0,\pi ]$

Ans: $f(x) = {\cos ^2}x + \sin x$

${f^\prime }(x) = 2\cos x( - \sin x) + \cos x$

$=  - 2\sin x\cos x + \cos x$

${f^\prime }(x) = 0{\text{ }}$

$\Rightarrow 2\sin x\cos x = \cos x \Rightarrow \cos x(2\sin x - 1) = 0$

$\Rightarrow \sin x = \dfrac{1}{2}$ or $\cos x = 0$

$\Rightarrow x = \dfrac{\pi }{6}$, or $\dfrac{\pi }{2}$ as $x \in [0,\pi ]$

$f\left( {\dfrac{\pi }{6}} \right) = {\cos ^2}\dfrac{\pi }{6} + \sin \dfrac{\pi }{6} = {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2} + \dfrac{1}{2} = \dfrac{5}{4}$

$f(0) = {\cos ^2}0 + \sin 0 = 1 + 0 = 1$

$f(\pi ) = {\cos ^2}\pi  + \sin \pi  = {( - 1)^2} + 0 = 1$

$f\left( {\dfrac{\pi }{2}} \right) = {\cos ^2}\dfrac{\pi }{2} + \sin \dfrac{\pi }{2} = 0 + 1 = 1$

absolute maximum value of $f$ is $\dfrac{5}{4}$ at $x = \dfrac{\pi }{6}$

The absolute minimum value of $f$ is 1 at $x = 0,x = \dfrac{\pi }{2}$, and $\pi $.

15. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $\mathrm{r}$ is $\dfrac{4 r}{3}$.

Ans: $V = \dfrac{1}{3}\pi {R^2}h$

$BC = \sqrt {{r^2} - {R^2}} $

$h = r + \sqrt {{r^2} - {R^2}} $

$\therefore V = \dfrac{1}{3}\pi {R^2}\left( {r + \sqrt {{r^2} - {R^2}} } \right) = \dfrac{1}{3}\pi {R^2}r + \dfrac{1}{3}\pi {R^2}\sqrt {{r^2} - {R^2}} $

$\therefore \dfrac{{dV}}{{dR}} = \dfrac{2}{3}\pi Rr + \dfrac{{2\pi }}{3}\pi R\sqrt {{r^2} - {R^2}}  + \dfrac{{{R^2}}}{3} \cdot \dfrac{{( - 2R)}}{{2\sqrt {{r^2} - {R^2}} }}$

$= \dfrac{2}{3}\pi Rr + \dfrac{{2\pi }}{3}\pi R\sqrt {{r^2} - {R^2}}  - \dfrac{{{R^3}}}{{3\sqrt {{r^2} - {R^2}} }}$

$= \dfrac{2}{3}\pi Rr + \dfrac{{2\pi Rr\left( {{r^2} - {R^2}} \right) - \pi {R^3}}}{{3\sqrt {{r^2} - {R^2}} }}$

$= \dfrac{2}{3}\pi Rr + \dfrac{{2\pi R{r^2} - 3\pi R{r^3}}}{{3\sqrt {{r^2} - {R^2}} }}$

$\dfrac{{dV}}{{d{R^2}}} = 0$

$\Rightarrow \dfrac{{2\pi rR}}{3} = \dfrac{{3\pi {R^2} - 2\pi R{r^2}}}{{3\sqrt {{r^2} - {R^2}} }}$

$\Rightarrow 2{\text{r}}\sqrt {{r^2} - {R^2}}  = 3{R^2} - 2{r^2}$

$\Rightarrow 4{r^2}\left( {{r^2} - {R^2}} \right) = {\left( {3{R^2} - 2{r^2}} \right)^2}$

$\Rightarrow 14{r^4} - 4{r^2}{R^2} = 9{R^4} + 4{r^4} - 12{R^2}{r^2}$

$\Rightarrow 9{R^4} - 8{r^2}{R^2} = 0$

$\Rightarrow 9{R^2} = 8{r^2}$

$\Rightarrow {R^2} = \dfrac{{8{r^2}}}{9}$

$\dfrac{{{d^2}V}}{{d{R^2}}} = \dfrac{{2\pi r}}{3} + \dfrac{{3\sqrt {{r^2} - {R^2}} \left( {2\pi {r^2} - 9\pi {R^2}} \right) - \left( {2\pi {R^3} - 3\pi {R^3}} \right)( - 6R)\dfrac{1}{{2{{\sqrt {{r^2} - R} }^2}}}}}{{9\left( {{r^2} - {R^2}} \right)}}$

$= \dfrac{{2\pi r}}{3} + \dfrac{{3\sqrt {{r^2} - {R^2}} \left( {2\pi {r^2} - 9\pi {R^2}} \right) - \left( {2\pi {R^3} - 3\pi {R^3}} \right)(3R)\dfrac{1}{{2{{\sqrt {{r^2} - R} }^2}}}}}{{9\left( {{r^2} - {R^2}} \right)}}$

when ${R^2} = \dfrac{{8{r^2}}}{9},\dfrac{{{d^2}V}}{{d{R^2}}} < 0$.

volume is the maximum when ${R^2} = \dfrac{{8{r^2}}}{9}$. ${R^2} = \dfrac{{8{r^2}}}{9}$, 

height of the cone $= r + \sqrt {{r^2} - \dfrac{{8{R^2}}}{9}}  = r + \sqrt {\dfrac{{{r^2}}}{9}}  = r + \dfrac{r}{3} = \dfrac{{4r}}{3}$

17. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $\dfrac{{2R}}{{\sqrt 3 }}$, also find the maximum volume.

Ans: $h = 2\sqrt {{R^2} - {r^2}} $

$V = \pi {r^2}h = 2\pi {r^2}\sqrt {{R^2} - {r^2}} $

$\therefore \dfrac{{dV}}{{dr}} = 4\pi r\sqrt {{R^2} - {r^2}}  + \dfrac{{2\pi {r^2}( - 2r)}}{{2\sqrt {{R^2} - {r^2}} }}$

$= 4\pi r\sqrt {{R^2} - {r^2}}  - \dfrac{{2\pi {r^3}}}{{\sqrt {{R^2} - {r^2}} }}$

$= \dfrac{{4\pi r\left( {{R^2} - {r^2}} \right) - 2\pi {r^3}}}{{\sqrt {{R^2} - {r^2}} }}$

$= \dfrac{{4\pi r{R^2} - 6\pi {r^3}}}{{\sqrt {{R^2} - {r^2}} }}$

Now, $\dfrac{{dV}}{{dr}} = 0 \Rightarrow 4\pi r{R^2} - 6\pi {r^3} = 0$

$\Rightarrow {r^2} = \dfrac{{2{R^2}}}{3}$

$\dfrac{{{d^2}V}}{{d{r^2}}} = \dfrac{{\sqrt {{R^2} - {r^2}} \left( {4\pi {R^2} - 18\pi {r^2}} \right) - \left( {4\pi r{R^2} - 6\pi {r^3}} \right)\dfrac{{( - 2r)}}{{2\sqrt {{R^2} - {r^2}} }}}}{{\left( {{R^2} - {r^2}} \right)}}$

$= \dfrac{{\left( {{R^2} - {r^2}} \right)\left( {4\pi {R^2} - 18\pi {r^2}} \right) + r\left( {4\pi r{R^2} - 6\pi {r^3}} \right)}}{{{{\left( {{R^2} - {r^2}} \right)}^{\dfrac{3}{2}}}}}$

$= \dfrac{{4\pi {R^4} - 22\pi {r^2}{R^2} + 12\pi {r^4} + 4\pi {r^2}{R^2}}}{{{{\left( {{R^2} - {r^2}} \right)}^{\dfrac{3}{2}}}}}$

${r^2} = \dfrac{{2{R^2}}}{3},\dfrac{{{d^2}V}}{{d{r^2}}} < 0.$

volume is maximum when ${r^2} = \dfrac{{2{R^2}}}{3}$. ${r^2} = \dfrac{{2{R^2}}}{3}$.

height of the cylinder is $2\sqrt {{R^2} - \dfrac{{2{R^2}}}{3}}  = 2\sqrt {\dfrac{{{R^2}}}{3}}  = \dfrac{{2R}}{{\sqrt 3 }}$.

volume of the cylinder is maximum when height of cylinder is $\dfrac{{2R}}{{\sqrt 3 }}$

18. Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height ${\text{h}}$ and semi veritical angle a is one-third that of the cone and the greatest volume of cylinder is $\dfrac{4}{{27}}\pi {h^2}{\tan ^2}a$.

Ans:

 $r = h\tan a$

since $\Delta AOG$ is similar to $\Delta CEG$,

$\dfrac{{AO}}{{OG}} = \dfrac{{CE}}{{EG}}$

$\Rightarrow \dfrac{h}{r} = \dfrac{H}{{r - R}}$

$\Rightarrow H = \dfrac{h}{r}(r - R) = \dfrac{h}{{h\tan a}}(h\tan a - R) = \dfrac{1}{{\tan a}}(h\tan a - R)$

volume of the cylinder is $V = \pi {R^2}H = \dfrac{{\pi {R^2}}}{{\tan a}}(h\tan a - R)$

$= \pi {R^2}h - \dfrac{{\pi {R^3}}}{{\tan a}}$

$\therefore \dfrac{{dV}}{{dR}} = 2\pi Rh\dfrac{{3\pi {R^2}}}{{\tan a}}$

$\dfrac{{dV}}{{dR}} = 0$

$\Rightarrow 2\pi Rh = \dfrac{{3\pi {R^2}}}{{\tan a}}$

$\Rightarrow 2h\tan a = 3R$

$\Rightarrow R = \dfrac{{2h}}{3}\tan a$

$\dfrac{{{d^2}V}}{{d{R^2}}} = 2\pi Rh - \dfrac{{6\pi R}}{{\tan a}}$

And, for $R = \dfrac{{2h}}{3}\tan a$, we have:

$\dfrac{{{d^2}V}}{{d{R^2}}} = 2\pi h - \dfrac{{6\pi }}{{\tan a}}\left( {\dfrac{{2h}}{3}\tan a} \right) = 2\pi h - 4\pi h =  - 2\pi h < 0$

volume of the cylinder is greatest when $R = \dfrac{{2h}}{3}\tan a$. $R = \dfrac{{2h}}{3}\tan a,H = \dfrac{1}{{\tan a}}\left( {h\tan a - \dfrac{{2h}}{3}\tan a} \right) = \dfrac{1}{{\tan a}}\left( {\dfrac{{h\tan a}}{3}} \right) = \dfrac{h}{3}$.

the maximum volume of cylinder can be obtained as

$\pi {\left( {\dfrac{{2h}}{3}\tan a} \right)^2}\left( {\dfrac{h}{3}} \right) = \pi \left( {\dfrac{{4{h^2}}}{9}{{\tan }^2}a} \right)\left( {\dfrac{h}{3}} \right) = \dfrac{4}{{27}}\pi {h^3}{\tan ^2}a$

19. A cylindrical tank of radius $10\;{\text{m}}$ is being filled with wheat at the rate of 314 cubic mere per hour. Then the depth of the wheat is increasing at the rate of

(A) $1\;{\text{m}}/{\text{h}}$ (B) $0.1\;{\text{m}}/{\text{h}}$ (C) $1.1\;{\text{m}}/{\text{h}}$

(D) $0.5\;{\text{m}}/{\text{h}}$

Ans: $V = \pi {({\text{ radius }})^2}{\text{x}}$ height

$= \pi {(10)^2}h\quad ($ radius $= 10\;{\text{m}})$

$= 100\pi h$

$\dfrac{{dV}}{{dt}} = 100\pi \dfrac{{dh}}{{dt}}$

Tank is being filled with wheat at rate of 314 cubic meters per hour.

 $\dfrac{{dV}}{{dt}} = 314\;{{\text{m}}^3}/{\text{h}}$

$314 = 100\pi \dfrac{{dh}}{{dt}}$

$\Rightarrow \dfrac{{dh}}{{dt}} = \dfrac{{314}}{{100(3.14)}} = \dfrac{{314}}{{314}} = 1$

The depth of wheat is increasing at $1\;{\text{m}}/{\text{h}}$.

The correct answer is ${\mathbf{A}}$.

20. The slope of the tangent to the curve $x = {t^2} + 3t - 8,y = 2{t^2} - 2t - 5$ at the point $(2, - 1)$ is

1. $\dfrac{{22}}{7}$

2. $\dfrac{6}{7}$ 

3. $\dfrac{7}{6}$

4. $\dfrac{{ - 6}}{7}$

Ans: $x = {t^2} + 3t - 8,{\kern 1pt} \,{\kern 1pt} {\text{and}}{\kern 1pt} \,y = 2{t^2} - 2t - 5$

$\therefore \dfrac{{dx}}{{dt}} = 2t + 3$ and $\dfrac{{dy}}{{dt}} = 4t - 2$

$\therefore \dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{dt}}a\dfrac{{dt}}{{dx}} = \dfrac{{4t - 2}}{{2t + 3}}$

given point is $(2, - 1)$.

At $x = 2$

${t^2} + 3t - 8 = 2$

$\Rightarrow {t^2} + 3t - 10 = 0$

$\Rightarrow (t - 2)(t + 5) = 0$

$\Rightarrow t = 2$ or ${\text{t}} =  - 5$

At $y =  - 1$, we have:

$2{t^2} - 2t - 5 =  - 1$

$\Rightarrow 2{t^2} - 2t - 4 = 0$

$\Rightarrow 2\left( {{t^2} - t - 2} \right) = 0$

$\Rightarrow (t - 2)(t + 1) = 0$

$\Rightarrow t = 2$ or $t =  - 1$

common value of $t$ is 2.

slope of tangent to given curve at point $(2, - 1)$ is

${\left. {\dfrac{{dy}}{{dx}}} \right]_{t = 2}} = \dfrac{{4(2) - 2}}{{2(2) + 3}} = \dfrac{{8 - 2}}{{4 + 3}} = \dfrac{6}{7}$

The correct answer is ${\mathbf{B}}$.

21. The line $y = mx + 1$ is tangent to the given curve ${y^2} = 4x$ if the value on $m$ is

1. $1\left( {} \right.$ B) $2\left( {} \right.$ C) $3($ D $)\dfrac{1}{2}$ 

Ans: Equation of the tangent to curve is $y = mx + 1$.

substituting $y = mx + 1$ in ${y^2} = 4x$,

$\Rightarrow {(mx + 1)^2} = 4x$

$\Rightarrow {m^2}{x^2} + 1 + 2mx - 4x = 0$

$\Rightarrow {m^2}{x^2} + x(2m - 4) + 1 = 0\quad $.....$(i)$

${(2m - 4)^2} - 4\left( {{m^2}} \right)(1) = 0$

$\Rightarrow 4{m^2} + 16 - 16m - 4{m^2} = 0$

$\Rightarrow 16 - 16m = 0$

$\Rightarrow m = 1$

the required value of $m$ is 1.

The correct answer is ${\mathbf{A}}$.

22. The normal at the point $(1,1)$ on the curve $2y + {x^2} = 3$ is

(A) $x + y = 0$ (B) $x - y = 0$ (C) $x + y + 1 = 0$ (D) $x - y = 1$

Ans: $\dfrac{{2dy}}{{dx}} + 2x = 0$

$\Rightarrow \dfrac{{dy}}{{dx}} =  - x$

${\left. {\therefore \dfrac{{dy}}{{dx}}} \right]_{(1,1)}} =  - 1$

slope of normal to curve at point $(1,1)$ is

$\dfrac{{ - 1}}{{{{\left. {\dfrac{{dy}}{{dx}}} \right]}_{(1,1)}}}} = 1$

equation of normal to given curve at $(1,1)$ is

$\Rightarrow y - 1 = 1(x - 1)$

$\Rightarrow y - 1 = x - 1$

$\Rightarrow x - y = 0$

The correct answer is ${\mathbf{B}}$.

23. The normal to the curve ${x^2} = 4y$ passing $(1,2)$ is

A) $x + y = 3({\mathbf{B}})x - y = 3({\mathbf{C}})x + y = 1({\mathbf{D}})x - y = 1$

Ans: $2x = 4{\mathbf{x}}\dfrac{{dy}}{{dx}}$

$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{x}{2}$

Slope of normal to curve at point $(h,k)$ is

$\dfrac{{ - 1}}{{{{\left. {\dfrac{{dy}}{{dx}}} \right]}_{(h,k)}}}} =  - \dfrac{2}{h}$

Equation of normal at point $(h,k)$ is

$y - k = \dfrac{{ - 2}}{h}(x - h)$

normal passes through the point $(1,2)$.

$2 - k = \dfrac{{ - 2}}{h}(1 - h)$ or $k = 2 + \dfrac{2}{h}(1 - h)\quad  \ldots ..(i)$

$(h,k)$ lies on the curves ${x^2} = 4y$, we have ${h^2} = 4k$.

$(h,k)$ lies on the curves ${x^2} = 4y$, we have ${h^2} = 4k$.

$\Rightarrow k = \dfrac{{{h^2}}}{4}$

$\dfrac{{{h^2}}}{4} = 2 + \dfrac{2}{h}(1 - h)$

$\Rightarrow \dfrac{{{h^3}}}{4} = 2h + 2 - 2h = 2$

$\Rightarrow {h^3} = 8$

$\Rightarrow h = 2$

$\therefore k = \dfrac{{{h^2}}}{4} \Rightarrow k = 1$

equation of normal is

$\Rightarrow y - 1 = \dfrac{{ - 2}}{2}(x - 2)$

$\Rightarrow y - 1 = (x - 2)$

$\Rightarrow x + y = 3$

The correct answer is ${\mathbf{A}}$.

24. The points on the curve $9{y^2} = {x^3}$, where the normal to the curve makes equal intercepts with the axes are

(A) $\left( {4, \pm \dfrac{8}{3}} \right)$ (B) $\left( {4,\dfrac{{ - 8}}{3}} \right)$ (C) $\left( {4, \pm \dfrac{3}{8}} \right)$

(D) $\left( { \pm 4,\dfrac{8}{3}} \right)$

Ans: $9(2y)\dfrac{{dy}}{{dx}} = 3{x^2}$

$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{{x^2}}}{{6y}}$

slope of normal to given curve at point $\left( {{x_1},{y_1}} \right)$ is $\dfrac{{ - 1}}{{dy}} =  - {\dfrac{{6{y_1}}}{{{x^2}}}_1}$

${\left. {\dfrac{{dy}}{{dx}}} \right]_{\left( {{x_1},{y_1}} \right)}}$

Equation of normal to curve at $\left( {{x_1},{y_1}} \right)$ is

$y - {y_1} = \dfrac{{ - 6{y_1}}}{{x_1^2}}\left( {x - {x_1}} \right).$

$\Rightarrow x_1^2y + x_1^2{y_1} = 6x{y_1} + 6{x_1}{y_1}$

$\Rightarrow 6{x_1}{y_1} + x_1^2y = 6{x_1}{y_1} + x_1^2{y_1}$

NCERT Solutions for Class 12 Maths – Free PDF Download

In today’s age, when students have so many assignments and tests to take, it becomes burdensome to handle them all. So, finding quick and quality solutions online would provide you with tremendous support in progressing with your studies. Finding NCERT Solutions for Class 12 Maths in PDF format on the official website of Vedantu would give you all the help that is needed in tackling all the problems of Chapter 6 Maths Class 12.

Important Points

• If x and y are two functions, the rate of change of x with respect to y is given as dy dx

• If y increases by increasing the value of x, then dy dx is positive.

• If y decreases by increasing the value of x, then dy dx is negative.

• If x and y are two variables with respect to t, another variable, then by Chain Rule dy dx=dy dt/dxdt provided dx dt ≠ 0. 

• At point (x0, y0), if

• dy dx

• = 0 then the equation of the normal is x = x0.

• At point (x0, y0), if dxdt

• = does not exist then the equation of the normal is y = y0.

Steps to evaluate Maxima and Minima

• Evaluate the points where f’(x) = 0.

• Get the end-points of the interval.

• Evaluate the value of the function (f) at all end-points.

• From the values obtained in step 3, you can easily find the maxima and minima. 

NCERT Solutions for Class 12 Maths PDF Download

• Chapter 1 - Relations and Functions

• Chapter 2 - Inverse Trigonometric Functions

• Chapter 3 - Matrices

• Chapter 4 - Determinants

• Chapter 5 - Continuity and Differentiability

• Chapter 6 - Application of Derivatives

• Chapter 7 - Integrals

• Chapter 8 - Application of Integrals

• Chapter 9 - Differential Equations

• Chapter 10 - Vector Algebra

• Chapter 11 - Three Dimensional Geometry

• Chapter 12 - Linear Programming

• Chapter 13 - Probability

NCERT Solution Class 12 Maths of Chapter 6 All Exercises

NCERT Solutions for Class 12 Maths Chapter 6

NCERT Solutions for Class 12 Maths Chapter 6 - Application of Derivatives are very useful resources for the students who are looking to practice NCERT questions based on every concept included in the chapter. Since Chapter 6 is quite an important one, it is only wise to solve as many questions as possible. Thus, students can easily download the PDF and practice the questions anywhere and anytime. 

Students can also download a plethora of other study materials including important questions and revision notes of all subjects.

6.1 Introduction

In Ch 6 Maths Class 12, the introduction part will be a recap of the last chapter where you got acquainted with concepts of derivative of a composite function, implicit functions, logarithmic functions, inverse trigonometric functions and exponential functions. 

In this chapter, you will learn how the application of derivatives works in different disciplines like social science, engineering, science, etc. This chapter will go through how to apply the application of derivatives to find out the rate of change of quantities.

Other Key Concepts Discussed in This Chapter Are

• Equations of a tangent and normal to a curve at a point.

• Find out turning points on the graph which will help you determine the points where maximum and minimum values of a function occur.

• Find intervals where a function increases and decreases.

• Approximations and errors.

The chapter will also give you ample practice with a variety of exercises of different kinds like short and long answers, objective types questions, etc. These activities would further strengthen all the important topics learned in this chapter and help you tremendously in your preparation for your board’s exams.

6.2 Rate of Change of Quantities

In NCERT Solutions for Class 12 Chapter 6 Applications of Derivatives, you would brush up all the learnings from the previous class. You would be reminded that to represent the rate of change of quantity with respect to another, we use the notation dy/dx, this means that y changes when x is changed.

You would further expand this concept to find out if changes in x and y are dependent on a 3rd variable (say z)  i.e. x= f(z) and y=g(z), Chain rule can be applied as shown below:

dy/dx = (dy/dz)/(dx/dz), given dx/dz <> =0

6.3 Increasing and Decreasing Functions

In this section of AOD Class 12 NCERT solutions, you would understand what is meant by increasing and decreasing function and how to determine whether a function is increasing or decreasing in a given range by using differentiation. You would define that a function is increasing in a range if the first derivative is positive in that range. Similarly, you would learn that a function is decreasing in a range if the derivative is negative in that range.

You will learn that in the case of trigonometric functions, one has to determine the quadrant in which they lie in order to know if they are increasing or decreasing. So, in this case, you would understand that for increasing the value of x, y is increasing then it is an increasing function, and if y decreases with an increase in the value of x, then it is a decreasing function. For example, sin x increases in the first quadrant while cos x is a decreasing function in the same quadrant. 

You should write down the above laws in the formula as depicted below

• Increasing Functions –> x1 < x2 in interval i implies f(x1) < f(x2) where x1 and x2 є i

• Decreasing Functions -> x1 < x2 in interval i implies f(x1) > f(x2) where x1 and x2 є i

6.4 Tangents and Normals

In this part of NCERT Solutions for Class 12 Maths Application of Derivatives, you would build your knowledge of the equation of straight lines learned in previous chapters. You would rekindle that the equation of a straight line on a curve y = f(x) with a finite slope s, passing through a given point (a, b) is given by:

y - b = s (x – a)

You would learn that a tangent to a point (a,b) on a curve is given by:

\[\frac {dy}{dx}\](a,b) and is denoted by f’(a) hence the equation of a tangent to the curve is given by : y – b = f’(a) (x – a). 

You would also get to know that normal is perpendicular to the tangent whose equation is given by y – b = (-1/ f’(a)) (x – a)

With Application of Derivatives Class 12 Solutions, you would get a good idea about what methods to adopt to calculate the expressions utilizing the slope formula.

6.5 Approximations

This part of Ch 6 Class 12 Maths NCERT Solutions will introduce you to the concept of using differentiation in order to find approximate values of certain quantities. You would get to know how approximations are useful when an exact numerical number is not easy to obtain. You would understand how to derive the approximate value of a quantity that changes in small measures by a change in another variable. So if y = f(x) and dx is a small increment in x which makes an increment of dy in the value of y, you would learn that dy = f (x + dx) – f(x) or dy = f’(x) dx.

The small change is generally denoted by Δ and approximation by ≈. So you will learn how to use these symbols to express the above equation as dy = (dy/dx) Δx.

From the learnings in this chapter, you will be able to figure out that the derivative of the dependent variable (y) is not equal to the increase in the variable while the derivative of the independent variable (x) is equal to the increase in the variable.

6.6 Maxima and Minima

This unit of NCERT Solutions for Class 12 Maths Chapter 6 describes the minimum and maximum values of a function in the form of derivatives. You will learn about turning points on the graph of a function where the graph reaches its highest point and the lowest point locally in a domain. You would understand how these points can be used to sketch the graph of a function.

You will also be able to find the absolute maximum and absolute minimum of a function which are useful in solving many applied problems. We would learn various definition around maxima and minima which are used for solving problems related to it like:

• In a range r, function f is said to have maximum value if there exists a point p in r where f(p) >= f(x) for all x є r. The value f(p) defines the maximum value of f in r, and the point p is termed as the point of the maximum value in r.

• In a range r, function f is said to have minimum value if there exists a point p in r where f(p) <= f(x) for all x є r. The value f(p) defines the maximum value of f in r, and the point p is termed as the point of the minimum value in r.

• In a range r, function f is said to have extreme value if there exists a point p in r where f(p) is either a maximum value or a minimum value of f in r and the point p is termed as the extreme point in r.

You would also learn how local maxima and local minima are represented and many theorems like the first derivative test, second derivative test around this concept.

Key Features of NCERT Solutions for Class 12 Maths Chapter 6

If you want to ace your exams and score well, having solutions to Maths problems becomes essential. Maths is an important subject for boards and also for many competitive exams. Hence if you get the solutions readily available by a stellar team of experts at Vedantu, you have the edge over others. The key benefits of Maths NCERT Solutions Class 12 Chapter 6 are:

• The solutions are well-crafted by subject matter experts who have given a proper explanation for every step of the problem.

• Using these solutions, you can get better clarity of the topic and be able to solve even difficult problems with proper time management.

• Many tips and tricks of remembering important formulas will be shared.

• You can always seek assistance from the teachers online in case you get doubts while going through the solutions.

Why should I Refer to Vedantu’s NCERT Solutions for Class 12 Maths Chapter 6?

There are various reasons why you should always opt for the NCERT Solutions for Maths Class 12 Chapter 6 created by Vedantu. They are:

• CBSE highly recommends the NCERT books to every class 12 student as these NCERT books proved to be the best guide with detailed study material including the important topics. Those appearing for the board examinations should opt for the NCERT Solutions for class 12 maths chapter 6 while solving the questions from the textbook.

• Our NCERT Solutions for Class 12 Maths Chapter 6 play a significant role for you as you can get to know the answers to all the questions in one single place. 

• NCERT Solutions for Class 12 maths Chapter 6 are prepared by our subject matter experts and faculties to help you in board exam preparations. These solutions help you to solve the problems easily and in a concise manner so that you score good marks in exams.

• NCERT solutions provide a detailed and step-by-step explanation of each answer to the questions asked in the exercises of the Class 12 Maths textbook Chapter 6.