x+y/2和2xy/x+y哪个大

Question

x+y/2和 2xy/x+y 哪个大

Answer 1

如果x,y都是正数
(x+y)/2-2xy/(x+y)
=[(x+y)^2-4xy]/(x+y)
=(x-y)^2/(x+y)≥0
所以
1.x=y时,(x+y)/2和2xy/(x+y)一样大
2.x≠y时,(x+y)/2大于2xy/(x+y)