Question

Given that `x,y,z` are positive reals such that `xyz=32` . The minimum value of `x^2+4xy+4y^2+2z^2` is ___________.

Answer 1

Correct Answer - 96
Using A.M `ge` GM for `x^(2), 2xy, 2xy, 4y^(2), z^(2), z^(2)`
`:. (x^(2) + 2xy + 2xy + 4y^(2) + z^(2) + z^(2))/(6) ge [16(xyz)^(4)]^(1//6)`
`=[16 (32)^(4)]^(1//6) = (2^(24))^(1//6) = 16`