JEE MAIN - Mathematics (2019 - 11th January Evening Slot - No. 1)
Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression $${{{x^m}{y^n}} \over {\left( {1 + {x^{2m}}} \right)\left( {1 + {y^{2n}}} \right)}}$$ is :
$${1 \over 2}$$
$${1 \over 4}$$
$${{m + n} \over {6mn}}$$
1
Explanation
$${{{x^m}{y^n}} \over {\left( {1 + {x^{2m}}} \right)\left( {1 + {y^{2n}}} \right)}} = {1 \over {\left( {{x^m} + {1 \over {{x^m}}}} \right)\left( {{y^n} + {1 \over {{y^n}}}} \right)}} \le {1 \over 4}$$
using AM $$ \ge $$ GM
using AM $$ \ge $$ GM
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