Let x, y be positive real numbers and m, n positive integers.

Question:

Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression

$\frac{x^{m} y^{n}}{\left(1+x^{2 m}\right)\left(1+y^{2 n}\right)}$ is :-

  1. $\frac{1}{2}$

  2. $\frac{1}{4}$

  3. $\frac{m+n}{6 m n}$

  4. 1


Correct Option: , 2

Solution:

$\frac{x^{m} y^{n}}{\left(1+x^{2 m}\right)\left(1+y^{2 n}\right)}=\frac{1}{\left(x^{m}+\frac{1}{x^{m}}\right)\left(y^{n}+\frac{1}{y^{n}}\right)} \leq \frac{1}{4}$

using $\mathrm{AM} \geq \mathrm{GM}$

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