设x>0,y>0,求证:1/2(x +y)2 +1/4(x +y)≥根号下xy(根号下x+根号下y)

问题描述:

设x>0,y>0,求证:1/2(x +y)2 +1/4(x +y)≥根号下xy(根号下x+根号下y)

原式=
1/2(x +y)(x+y+1/2) (提一个X+Y)
≥根号下xy(x+1/4+y+1/4)
≥根号下xy(根号下x+根号下y)
用两次均值不等式