If `alpha+beta, gamma epsilon R^(+)` such that `alpha gamma =1/(beta)` then maximum value of `alpha^(beta+gamma), .beta^(beta+gamma).gamma^(alpha+beta)` is_____

Answer

1

Explanation

`beta=1/(alpha gamma)`
Assume `alpha ge beta ge gamma` (say)
`impliesalpha ge 1` and `gamma le1`
`:.(alpha^(beta+gamma).gamma^(alpha+beta))/((alpha gamma)^(alpha+gamma))=(gamma^(beta-gamma))/(alpha^(alpha-beta)) le 1`