x,y,z∈(0,1),且x+y+z=2,求证1

题目

x,y,z∈(0,1),且x+y+z=2,求证1

答案

先证右边.易知任意实数x,y,z都有x²+y²+z²≥xy+yz+zx.(x+y+z)²=x²+y²+z²+2(xy+yz+zx)≥3(xy+yz+zx)∴xy+yz+zx≤(x+y+z)²/3=4/3.再证左边.xy+yz+zx=xy+(x+y)(2-x-y)=-x²...