If a>b>0 thenthe maximum value of (a b(a^2-b^2) sinx cosx)/(a^2 sin^2 x+b^2 cos^2 x) in (0, (π)/

Name: Problem 20, Chapter 7: Differential Calculus for JEE Main and Advanced
Uploaded: 2021-11-24T08:34:27-08:00
Duration: 1 min 6 s
Channel: Hast Aggarwal
Description: If a>b>0 thenthe maximum value of (a b(a^2-b^2) sinx cosx)/(a^2 sin^2 x+b^2 cos^2 x) in (0, (π)/(2)) is (A) a^2-b^2 (B) (a^2-b^2)/(2) (C) (a^2+b^2)/(2) (D) none of these

step 1 So in this question we have y is equal to a b a square minus b square sine 2 x divided by size o

Question

If $a>b>0$ thenthe maximum value of $\frac{a b\left(a^{2}-b^{2}\right) \sin x \cos x}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}$ in $\left(0, \frac{\pi}{2}\right)$ is (A) $a^{2}-b^{2}$ (B) $\frac{a^{2}-b^{2}}{2}$ (C) $\frac{a^{2}+b^{2}}{2}$ (D) none of these

   If $a>b>0$ thenthe maximum value of $\frac{a b\left(a^{2}-b^{2}\right) \sin x \cos x}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}$ in $\left(0, \frac{\pi}{2}\right)$ is
(A) $a^{2}-b^{2}$
(B) $\frac{a^{2}-b^{2}}{2}$
(C) $\frac{a^{2}+b^{2}}{2}$
(D) none of these

Differential Calculus for JEE Main and Advanced

Vinay Kumar 3rd Edition

Chapter 7, Problem 20

Instant Answer

Solved by Expert Hast Aggarwal

11/24/2021

Step 1

Step 1: We are given the function \[y = \frac{a b\left(a^{2}-b^{2}\right) \sin x \cos x}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}\] We can simplify this by using the identity $\sin 2x = 2\sin x \cos x$ to get \[y = \frac{a b\left(a^{2}-b^{2}\right) \sin 2x}{2(a^{2} Show more…

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If $a>b>0$ thenthe maximum value of $\frac{a b\left(a^{2}-b^{2}\right) \sin x \cos x}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}$ in $\left(0, \frac{\pi}{2}\right)$ is (A) $a^{2}-b^{2}$ (B) $\frac{a^{2}-b^{2}}{2}$ (C) $\frac{a^{2}+b^{2}}{2}$ (D) none of these

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Key Concepts

Trigonometric Identities

Trigonometric identities, such as the double-angle formulas, are essential tools for simplifying expressions involving sine and cosine. These formulas allow the transformation of products into single trigonometric functions, which can make the process of optimization more manageable.

Function Optimization

Function optimization is a core concept in calculus and algebra, focusing on finding the maximum or minimum values of a given function. This generally involves techniques like differentiation to identify critical points, or transforming the function to reveal its extremal behavior directly.

Substitution and Transformation Techniques

Substitution and transformation techniques involve rewriting a function in a simpler or more insightful form by introducing new variables or applying algebraic identities. These methods help in reducing the complexity of an expression, thereby facilitating the process of finding its optimum values.

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