已知：x,y为互不相等的正数，试比较x五次方+y五次方和x四次方y+xy四次方的大小？

x^5+y^5-(x^4y+xy^4)=x^4-x^4y+y^5-xy^4=x^4(x-y)+y^4(y-x)
=(x-y)(x^4-y^4)=(x-y)(x^2+y^2)(x^2-y^2)
=(x-y)(x^2+y^2)(x+y)(x-y)
=(x-y)^2(x+y)(x^2+y^2)＞0
所以x^5+y^5＞x^4y+xy^4

另一个方法(均值不等式法)
x^5+x^5+x^5+x^5+y^5≥5[(x^5)^4*y^5]^(1/5)=5x^4y
同理4y^5+x^5≥5xy^4(当x=y时成立)
两式相加可得x^5+y^5＞x^4y+xy^4 (因x≠y)