求证(x+y)(x^3+y^3)大于等于0

(x+y)(x^3+y^3)
=(x+y)(x+y)(x^2-xy+y^2)
=((x+y)^2)((x-y/2)^2+(3y^2)/4)>=0
因为(x+y)^2>=0，(x-y/2)^2+(3y^2)/4>=0

当X>0,Y>0时 X+Y>0 X^3+Y^3>0 (X+Y)(X^3+Y^3)>0
当X<0,Y<0时 X+Y<0 X^3+Y^3<0 (X+Y)(X^3+Y^3)>0
当X=0,Y=0时 X+Y=0 X^3+Y^3=0 (X+Y)(X^3+Y^3)=0
所以(x+y)(x^3+y^3)大于等于0

(x+y)2(x2+y2-xy)=(x+y)2(2x2+2y2-2xy)/2=(x+y)2(x2+y2+(x-y)2)/2
因为(x+y)2,x2+y2+(x-y)2大于或等于零,所以原式大于或等于零