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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`

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The correct Answer is:
C
Consider, `(x^(m)y^(n))/((1 + x^(2m))(1 + y^(2n)))`
`= (1)/((x^(m) + x^(-m)) (1 + y^(2n)))`
By using AM `ge` GM (because x, `y in R^(+)`), we get
`(x^(m) + x^(-m)) ge 2 and (y^(n) + y^(-n)) ge2`
`[ :' " if " x gt 0, " then " x + (1)/(x) ge 2]`
`rArr (x^(m) + x^(-m)) (y^(n) + y^(-n)) ge 4`
`rArr (1)/((x^(m) + x^(-m)) (y^(n) + y^(-n))) le (1)/(4)`
`:.` Maximum value `= (1)/(4)`
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