   Question Answers

# If the function $f(x)=\dfrac{K\sin x+2\cos x}{\sin x+\cos x}$ is strictly increasing for all values of x, then(A). K<1(B). K>1(C). K<2(D). K>2  HINT: - For finding whether a function is increasing or decreasing at a particular value of x, we follow the following steps and get to the answer
1. First of all, we take the derivative of the function that is given to us.
2. Then we put the value of x at which we have to find whether the function is increasing or decreasing.
3. If the value of the derivative of the function at the entered value of x comes out to be positive, then we can say that the function is increasing and if the value of the derivative of the function at the entered value of x comes out to be negative, then we can say that the function is decreasing.

Complete step-by-step solution -
The most important formulae that would be used in solving this question are as follows
1. Quotient rule of differentiation:-
$\dfrac{d\left( \dfrac{u}{v} \right)}{dx}=\dfrac{v\cdot \dfrac{du}{dx}-u\cdot \dfrac{dv}{dx}}{{{v}^{2}}}$
2. ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ .
As mentioned in the question, we have to find the value of K for which the function that is given in the question is increasing for all values of x.
Now, as mentioned in the question, we have to first take the derivative of the function as follows
\begin{align} & f(x)=\dfrac{K\sin x+2\cos x}{\sin x+\cos x} \\ & {{\left( \dfrac{u}{v} \right)}^{\prime }}=\dfrac{d\left( \dfrac{u}{v} \right)}{dx}=\dfrac{v\cdot \dfrac{du}{dx}-u\cdot \dfrac{dv}{dx}}{{{v}^{2}}} \\ & {f}'(x)=\dfrac{\left( K\cos x-2\sin x \right)\left( \sin x+\cos x \right)-\left( \cos x-\sin x \right)\left( K\sin x+2\cos x \right)}{{{\left( \sin x+\cos x \right)}^{2}}} \\ & {f}'(x)=\dfrac{K\sin x\cdot \cos x-2{{\sin }^{2}}x+K{{\cos }^{2}}x-2\sin x\cdot \cos x-K\sin x\cdot \cos x-2{{\cos }^{2}}x+K{{\sin }^{2}}x+2\sin x\cdot \cos x}{{{\left( \sin x+\cos x \right)}^{2}}} \\ & {f}'(x)=\dfrac{-2{{\sin }^{2}}x+K{{\cos }^{2}}x-2{{\cos }^{2}}x+K{{\sin }^{2}}x}{{{\left( \sin x+\cos x \right)}^{2}}} \\ & {f}'(x)=\dfrac{K\left( {{\sin }^{2}}x+{{\cos }^{2}}x \right)-2\left( {{\sin }^{2}}x+{{\cos }^{2}}x \right)}{{{\left( \sin x+\cos x \right)}^{2}}} \\ & {f}'(x)=\dfrac{K-2}{{{\left( \sin x+\cos x \right)}^{2}}} \\ \end{align}
(Using the identities that are mentioned in the hint that is ${{\sin }^{2}}x+{{\cos }^{2}}x=1$)
Now, in the above expression, for f(x) to be increasing for all values of x, then the derivative of the function should be positive.
As the denominator is a square of a number, so, it cannot be negative; hence, we can say that we have to make only the numerator positive and for that we can write as follows
\begin{align} & \dfrac{K-2}{{{\left( \sin x+\cos x \right)}^{2}}}>0 \\ & K-2>0 \\ & K>2 \\ \end{align}
Hence, we have to get the value of K as mentioned above.

NOTE: - The students can make an error if they don’t know what the principle is behind the derivative test for finding the increasing or decreasing of a function and the principle that is involved here is that on taking the derivative, we are actually finding the slope of the function at a particular value of x when that value of x is entered into the function’s derivative.

Electromagnetic Spectrum X-rays  Analytic Function  One to One Function  Kidney Function Test  What Happens if the Earth Stops Rotating?  Cos 360  CBSE Class 12 Maths Formulas  Sin Cos Formula  CBSE Class 12 Maths Chapter-12 Linear Programming Formula  CBSE Class 6 Maths Chapter 12 - Ratio and Proportion Formulas  Important Questions for CBSE Class 12 Chemistry Chapter 8 - The d and f Block Elements  Important Questions for CBSE Class 12 Macro Economics Chapter 5 - Government Budget and the Economy  Important Questions for CBSE Class 12 Chemistry Chapter 1 - The Solid State  Important Questions for CBSE Class 12 Maths Chapter 12 - Linear Programming  Important Questions for CBSE Class 8 Maths Chapter 12 - Exponents and Powers  Important Questions for CBSE Class 12 Maths Chapter 1 - Relations and Functions  Important Questions for CBSE Class 12 Maths Chapter 5 - Continuity and Differentiability  Important Questions for CBSE Class 12 Chemistry Chapter 7 - The p-Block Elements  Important Questions for CBSE Class 6 Maths Chapter 12 - Ratio and Proportion  Important Questions for CBSE Class 12 Maths Chapter 7 - Integrals  CBSE Class 12 Maths Question Paper 2020  Maths Question Paper for CBSE Class 12 - 2013  Previous Year Question Paper for CBSE Class 12 Maths - 2014  CBSE Previous Year Question Papers Class 12 Maths with Solutions  Maths Question Paper for CBSE Class 12 - 2016 Set 1 C  Maths Question Paper for CBSE Class 12 - 2016 Set 1 E  Maths Question Paper for CBSE Class 12 - 2016 Set 1 S  CBSE Class 12 Maths Question Paper 2018 with Solutions - Free PDF  CBSE Class 12 Maths Question Paper 2015 with Solutions - Free PDF  Maths Question Paper for CBSE Class 12 - 2016 Set 1 N  RD Sharma Class 12 Maths Solutions Chapter 17 - Increasing and Decreasing Functions  RD Sharma Class 12 Solutions Chapter 17 - Increasing and Decreasing Functions (Ex 17.2) Exercise 17.2  RD Sharma Class 12 Solutions Chapter 17 - Increasing and Decreasing Functions (Ex 17.1) Exercise 17.1  RD Sharma Class 12 Maths Solutions Chapter 29 - The Plane  Textbooks Solutions for CBSE & ICSE Board of Class 6 to 12 Maths & Science  RS Aggarwal Class 12 Solutions Chapter-28 The Plane  NCERT Solutions for Class 9 English Beehive Chapter 11 - If I Were You  RD Sharma Class 12 Solutions Chapter 12 - Higher Order Derivatives (Ex 12.1) Exercise 12.1  NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming (Ex 12.1) Exercise 12.1  NCERT Solutions Class 11 English Woven Words Poem Chapter 12 Ajamil and the Tigers  