Answer

Verified

381.6k+ views

**Hint:**We will solve this in a simpler way. Here the interval is closed that means for f(x) = 13 the value of x can neither be less than 1 nor be greater than 2. And we will check this by putting values of x as 1 and 2 in the given function and also by using the mean value theorem.

**Complete step-by-step answer:**

Step-1

We have \[f(x) = {x^3} + \cos \pi x + 7\]

In the interval of [1,2],

Step-2

$f(1) = {1^3} + \cos \pi + 7$

$ \Rightarrow f(1) = 1 + 1 + 7$

$ \Rightarrow f(1) = 9$……………(1)

And $f(2) = {2^3} + \cos 2\pi + 7$

Or, $f(2) = 8 + 1 + 7$

Or, $f(2) = 16$………..(2)

Step-3

From above two value we get to know that $f(1) < f(x) = 13 < f(2)$

So, from this we get to know that \[f(x) = 13\] has a solution in the interval of (1,2).

Step-4

For\[f(x) = {x^3} + \cos \pi x + 7\] to be continuous f’(x) must be greater than 0.

For that,

$f(c) = \dfrac{{f(2) - f(1)}}{{2 - 1}}$

$f(c) = \dfrac{{16 - 9}}{{2 - 1}}$

$f(c) = 7$

Step-5

Again differentiating f(x) we get,

$f'(x) = 3{x^2} - \pi \sin \pi x$

Step-6

We know that,

$f'(x) = 3{x^2} - \pi \sin \pi x = f(c)$

Or, $f'(x) = 3{x^2} - \pi \sin \pi x = 7$

Or, $f'(x) = 3{x^2} - \pi \sin \pi x > 0$, for all x belongs to closed [1,2]

Step-7

Hence, f(x) is a continuous function at [1,2].

F(x) is a strictly increasing function.

Therefore for \[f(x) = 13\], it has exactly one solution in the closed interval of [1,2].

**So, the correct answer is “Option A”.**

**Note:**Mean value theorem- The mean value theorem states that if a function f is continuous on the closed interval of [a, b ] and differentiable on the open interval ( a, b ), then there exist a point C in the interval (a, b ) such that f’(c) is equal to the functions average rate of change over closed interval [a, b ].

Difference between open interval and closed interval is, an open interval does not include its limit points while closed interval does.

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Why is the adrenaline hormone called fight or flight class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between lanthanoids and actinoids class 12 chemistry CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Give 10 examples of unisexual and bisexual flowers

Open circulatory system is present in I Arthropods class 12 biology CBSE

Name the highest peak of the Indian Himalayas class 8 social science CBSE