已知函数X≥0,Y≥0,求证X（5次方）+Y（5次方）≥X³Y²+X²Y³.._数学解答

已知函数X≥0,Y≥0,求证X（5次方）+Y（5次方）≥X³Y²+X²Y³..

人气：465 ℃　时间：2019-11-12 06:26:03

解答

x^5+y^5-x^3y^2-x^2y^3=x^3(x^2-y^2)-y^3(x^2-y^2)=(x^2-y^2)(x^3-y^3)=(x+y)(x-y)(x-y)(x^2+xy+y^2)=(x-y)^2(x+y)(x^2+y^2+xy)因为x>=0,y>=0所以x+y>=0 x^2+y^2+xy>=0 (x-y)^2>=0所以x^5+y^5-x^3y^2-x^2y^3>=0即x^5...