Let x, y be positive real numbers and m, n be | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) We are given the expression $E = \frac{x^m y^n}{(1 + x^{2m})(1 + y^{2n})}$.We can rewrite this expression as $E = \frac{x^m}{1 + x^{2m}} 	imes \f

Vedclass · Accepted Answer

$\frac{1}{4}$

Vedclass · Answer

(C) We are given the expression $E = \frac{x^m y^n}{(1 + x^{2m})(1 + y^{2n})}$.We can rewrite this expression as $E = \frac{x^m}{1 + x^{2m}} 	imes \frac{y^n}{1 + y^{2n}}$.Dividing the numerator and denominator of each fraction by $x^m$ and $y^n$ respectively,we get:$E = \frac{1}{\left( \frac{1}{x^m} + x^m ight)} 	imes \frac{1}{\left( \frac{1}{y^n} + y^n ight)}$.By the Arithmetic Mean-Geometric Mean ($AM$-$GM$) inequality,for any positive real number $a$,$a + \frac{1}{a} \ge 2$,with equality holding when $a = 1$.Therefore,$x^m + \frac{1}{x^m} \ge 2$ and $y^n + \frac{1}{y^n} \ge 2$.This implies that $\frac{1}{x^m + \frac{1}{x^m}} \le \frac{1}{2}$ and $\frac{1}{y^n + \frac{1}{y^n}} \le \frac{1}{2}$.Multiplying these inequalities,we get the maximum value of $E$ as $\frac{1}{2} 	imes \frac{1}{2} = \frac{1}{4}$.

Let $x, y$ be positive real numbers and $m, n$ be positive integers. The maximum value of the expression $\frac{x^m y^n}{(1 + x^{2m})(1 + y^{2n})}$ is

Similar Questions

The extreme values of $1 + 4 \sin \theta + 3 \cos \theta$ are

If $A + B + C = \pi ,$ then $\tan ^2 \frac{A}{2} + \tan ^2 \frac{B}{2} + \tan ^2 \frac{C}{2}$ is always

Prove that $\cos ^{2} x+\cos ^{2}\left(x+\frac{\pi}{3}\right)+\cos ^{2}\left(x-\frac{\pi}{3}\right)=\frac{3}{2}$

Let $a$ be the maximum value of $(3 \cos \theta - 4 \sin \theta)$ and $\theta \neq \frac{n \pi}{2}$. If $\alpha = a \sin^2 \theta \cos^3 \theta$ and $\beta = a \sin^3 \theta \cos^2 \theta$,then $\sqrt{\frac{(\alpha^2 + \beta^2)^5}{(\alpha \beta)^4}} = $

If the equation $\frac{x^2 + 5}{2} = x - 2\cos(ax + b)$ has at least one solution,then $(b + a)$ can be equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module