What is the minimum value of (a^2)/( ^2 x)+(b^2)/( ^2 x) where a > 0 and b > 0 ?

Correct answer: A. (a+b)^2 For minimum value aligned & (a^2)/( ^2 x)+(b^2)/( ^2 x) & =a^2 ^2 x+b^2 cosec^2 x & =a^2(1+ ^2 x)+b^2(1+ ^2 x) & =a^2+b^2+Q^2 ^2 x+b^2 ^2 x & a^2 ^2 x+b^2 ^2 x ≥ 2 a b & minimum value =a^2+b^2+2 a b & =(a+b)^2 aligned

NDA 2019

Exam: NDA
Year: 2019
Subject: Mathematics
Chapter: Application of Derivatives

NDA 2019 · Mathematics · Application of Derivatives

What is the minimum value of (a^2)/( ^2 x)+(b^2)/( ^2 x) where a >...

What is the minimum value of

\frac{a^2}{\cos ^2 x}+\frac{b^2}{\sin ^2 x}

where

a > 0

and

b > 0

Options

A $(a+b)^2$
B $(a-b)^2$
C $a^2+b^2$
D $\left|a^2+b^2\right|$

Correct answer

A. $(a+b)^2$

Step-by-step solution

For minimum value

\begin{aligned} & \frac{a^2}{\cos ^2 x}+\frac{b^2}{\sin ^2 x} \\ & =a^2 \sec ^2 x+b^2 \operatorname{cosec}^2 x \\ & =a^2\left(1+\tan ^2 x\right)+b^2\left(1+\cot ^2 x\right) \\ & =a^2+b^2+Q^2 \tan ^2 x+b^2 \cot ^2 x \\ & \because a^2 \tan ^2 x+b^2 \cot ^2 x \geq 2 a b \\ & \therefore \text{ minimum value }=a^2+b^2+2 a b \\ & =(a+b)^2 \end{aligned}

Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

Passage: Let S and T be the sets where f(x)=(x^3)/(3)-(5x^2)/(2)+6x+7 decreases and… NDA 2026 What is the length of the longest interval in which the function f(x)=2 ^2 x-1 is… NDA 2026 Passage: Let S and T be the sets where f(x)=(x^3)/(3)-(5x^2)/(2)+6x+7 decreases and… NDA 2026 For the following three (03) items: Let y = f(x) = x ^ -1 x 1 - x^2 + 1 - x^2 . What is… NDA 2025 Consider the following statements regarding the function f(x) = (1)/(x - 5)… NDA 2025 Consider the following for the two (02) items that follow: Let ABC be a triangle… NDA 2025 Consider the following for the two (02) items that follow: Let ABC be a triangle… NDA 2025 A wire of length 20 cm is to be bent into a rectangle. Which of the following statements… NDA 2025 Consider the following for the two (02) items that follow: Let the function f(x) = x^2 +… NDA 2025 For the following two (02) items: Consider the function f(x) = 1 - [3] (x-1)^2 . The… NDA 2025 Consider the following for the two (02) items that follow: The slope of the tangent to… NDA 2025 For the following three (03) items: Consider the function f(x) = x|x| . Consider the… NDA 2025 Consider the following statements: Statement-I: The function f(x) = (x^3 + 128)/(x) has a… NDA 2025 Paragraph: Let f(x)= 2 x+x on [(-π)/(2), (π)/(2)] Question: What is the greatest value of… NDA 2024 Let f(x)=(x)/( x) ;(x > 1) Consider the following statements: 1. f(x) is increasing in… NDA 2024 Paragraph: Let f(x)= 2 x+x on [(-π)/(2), (π)/(2)] Question: What is the least value of… NDA 2024 Full Application of Derivatives list Application of Derivatives JEE Maths PYQs JEE Mathematics PYQs All NDA PYQs