Under which one of the following conditions does the function f(x)=(p x)^2+(q cosec x)^2 attain minimum value?

Correct answer: A. ^2 x=(q)/(p) aligned & f(x)=(P x)^2+(q cosec x)^2 & f^ (x)=2 P^2 ^2 x x-2 q^2 cosec^2 x · x=0 & (p^2)/(q^2)= cosec^2 x · x ^2 x · x = ^4 x & ^2 x=(p)/(q) ^2 x=(q)/(p) & f^ (x)=2 p^2( x+ ^3 x)-2 q^2( x+ ^3 x) & f^ (x)=2 p^2 ^2 x(1+3 ^2 x)+2 q^2 cosec^2 x & (1+3 ^2 x) & For ^2 x=(q)/(p) & f^ (x) > 0 & f(x) is minimum when ^2 x=(q)/(p) aligned

NDA 2022

Exam: NDA
Year: 2022
Subject: Mathematics
Chapter: Application of Derivatives

NDA 2022 · Mathematics · Application of Derivatives

Under which one of the following conditions does the function...

Under which one of the following conditions does the function

f(x)=(p \sec x)^2+(q \operatorname{cosec} x)^2

attain minimum value?

Options

A $\tan ^2 x=\frac{q}{p}$
B $\cot ^2 x=\frac{q}{p}$
C $\tan ^2 x=p q$
D $\cot ^2 x=p q$

Correct answer

A. $\tan ^2 x=\frac{q}{p}$

Step-by-step solution

\begin{aligned} & f(x)=(P \sec x)^2+(q \operatorname{cosec} x)^2 \\ & f^{\prime}(x)=2 P^2 \sec ^2 x \tan x-2 q^2 \operatorname{cosec}^2 x \cdot \cot x=0 \\ & \Rightarrow \frac{p^2}{q^2}=\frac{\operatorname{cosec}^2 x \cdot \cot x}{\sec ^2 x \cdot \tan x}=\cot ^4 x \\ & \Rightarrow \cot ^2 x=\frac{p}{q} \Rightarrow \tan ^2 x=\frac{q}{p} \\ & f^{\prime}(x)=2 p^2\left(\tan x+\tan ^3 x\right)-2 q^2\left(\cot x+\cot ^3 x\right) \\ & f^{\prime \prime}(x)=2 p^2 \sec ^2 x\left(1+3 \tan ^2 x\right)+2 q^2 \operatorname{cosec}^2 x \\ & \left(1+3 \cot ^2 x\right) \\ & \text{For } \tan ^2 x=\frac{q}{p} \\ & f^{\prime \prime}(x) > 0 \\ & \therefore \quad f(x) \text{ is minimum when } \tan ^2 x=\frac{q}{p}\end{aligned}

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