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If x and y are positive real numbers an...

If `x and y` are positive real numbers and `m, n` are any positive integers, then prove that `(x^n y^m)/((1+x^(2n))(1+y^(2m))) lt 1/4`

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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