Ques.-4

Algebra Level 3

If x x and y y are positive real numbers and m , n m,n are positive integers, then what is the maximum value of x m ⋅ y n ( 1 + x 2 m ) ⋅ ( 1 + y 2 n ) \dfrac{x^m \cdot y^n}{(1+x^{2m}) \cdot (1+y^{2n})}

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2 1 2 \frac{1}{2} 1 4 \frac{1}{4} 4

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Sandeep Rathod
Feb 25, 2015

x m ⋅ y n ( 1 + x 2 m ) ⋅ ( 1 + y 2 n ) \dfrac{x^m \cdot y^n}{(1+x^{2m}) \cdot (1+y^{2n})}

1 ( x m + 1 x m ) ( y n + 1 y n ) \dfrac{1}{( x^m + \dfrac{1}{x^m})(y^{n} + \dfrac{1}{y^{n}})}

Since x and y are positive reals so applying A.M G.M inequality we get 0.25

We want maximum value so denominator is to be minimum , thus 1/4

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