若x>y>z>0,求证：x-z+27/[(x+z)y-xz-y^2]≥9

Question

若x>y>z>0,求证：x-z+27/[(x+z)y-xz-y^2]≥9

Answer 1

左边分母进行化简
(x+z)y-xz-y^2
=xy+zy-xz-y^2
=y(x-y)+z(y-x)
=(x-y)(y-z)
所以左边=(x-y)+(y-z)+ 27/(x-y)(y-z)
用均值不等式
≥9