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 :-
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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