高二 不等式

(x^3+y^3)-(x^2y+xy^2)=(x+y)(x^2-xy+y^2)-xy(x+y)=(x+y)(x^2-2xy+y^2)=(x+y)(x-y)^2x＞0，y＞０所以x+y>0,(x-y)^2≥0(x+y)(x-y)^2≥0所以x^3+y^3≥x^2y+xy^2

x^3+y^3=（x+y）*（x^2-xy+y^2）x^2y+xy^2=xy(x+y)(x+y）*（x^2-xy+y^2）-xy(x+y)=(x+y)*(x^2-xy+y^2-xy)=(x+y)*(x-y)^2因为x＞，y＞０，所以(x+y)*(x-y)^2大于0，即x^3+y^3大于x^2y+xy^2