# Let x,y be positive real numbers and m,n positive integers. The maximum value of the expression $\frac{x^{m}y^{n}}{(1+x^{2m})(1+y^{2n})}$  is :Option 1)  1Option 2)$\frac{1}{2}$Option 3)  $\frac{1}{4}$Option 4)  $\frac{m+n}{6mn}$

Relation between AM, GM and HM of two positive numbers -

$AM\geqslant GM\geqslant HM$

- wherein

Inequality of the three given means.

1

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

Option 1)

1

Option 2)

$\frac{1}{2}$

Option 3)

$\frac{1}{4}$

Option 4)

$\frac{m+n}{6mn}$

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