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

Let x, y be positive real numbers and m, n be positive integers, The maximum value of the expression
`(x^(m)y^(n))/((1+x^(2m))(1+y^(2n)))` is

A

`1/4`

B

`1`

C

`1/2`

D

`(m +n)/(6mn)`

Text Solution

Verified by Experts

The correct Answer is:
A
The given expression can be written as `(1)/((x ^(m)+(1)/(x ^(m)))(y ^(n) + (1)/(y ^(n))))`
Now, `x ^(m) +(1)/(x ^(m))ge 2 and y ^(n) +(1)/(y ^(n))ge 2 implies (x ^(m) + (1)/(x ^(m))) (y ^(n) +(1)/(y ^(n)))ge4`
`therefore` Maximum value of `(1)/((x ^(m)+ (1)/(x ^(n)))(y ^(n) + (1)/(y ^(n))))is 1/4`
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