证明：对任意实数x,y,有x4+y4>=0.5xy（x+y）2

题目

证明：对任意实数x,y,有x4+y4>=0.5xy（x+y）2
4和2是指四次方和平方

答案

2x^4+2y^4-xy(x+y)^2= 2x^4+2y^4-xy(x^2+2xy+y^2)= 2x^4+2y^4-x^3y-2(xy)^2-xy^3= x^4+y^4-2(xy)^2+x^4+y^4-x^3y-xy^3= (x^2-y^2)^2+x^3(x-y)-y^3(x-y)= (x^2-y^2)^2+(x-y)(x^3-y^3)= (x^2-y^2)^2+(x-y)^2(x^2+xy+y^...