已知xy为任意实数,求证x^4+y^4≥1/2xy(x+y)^2
问题描述：
已知xy为任意实数,求证x^4+y^4≥1/2xy(x+y)^2
答
2(x^4+y^4)-xy(x+y)^2=2x^4+2y^4-x^3y-2x^2y^2-xy^3=(x^4-2x^2y^2+y^4)+(x^4+y^4-x^3y-xy^3)=(x^2-y^2)^2+x^3(x-y)+y^3(y-x)=(x^2-y^2)^2+(x-y)(x^3-y^3)=(x^2-y^2)^2+(x-y)[(x-y)(x^2+xy+y^2)]=(x^2-y^2)^2+(x-y)^2...