初二暑假数学作业中的一道题……

x^5+y^5-x^4y-xy^4
=x^4(x-y)-y^4(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)
=(x^2+y^2)(x+y)(x-y)^2
x，y为互不相等的正数
x^2+y^2>0
x+y>0
x-y不等于0
所以(x-y)^2>0
所以(x^2+y^2)(x+y)(x-y)^2>0
所以x^5+y^5-x^4y-xy^4〉0
所以x^5+y^5>x^4y+xy^4

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

x，y为互不相等的正数
x^2+y^2>0,x+y>0,(x-y)^2>0
所以x^5+y^5-(x^4y+xy^4)>0
所以
x^5+y^5>x^4y+xy^4

x^5+y^5较大 用作差法
原式=x^5+y^5-x^4y-y^4x
=x^4(x-y)+y^4(y-x)
=x^4(x-y)-y^4(x-y)
=(x^4-y^4)(x-y)
若x大于y
因为x y为正数
所以(x^4-y^4)大于0
(x-y)大于0
所以x^5+y^5较大
若x^4y+y^4x较大
则x^4-y^4小于0
x-y小于0
所以=(x^4-y^4)(x-y)大于0
所以x^4+y^4较大
综上所述x^4+y^4较大

(x^5+y^5)-(x^4y+y^4x)
=x^4*(x-y)+y^4*(y-x)
=(x^4-y^4)*(x-y)
=(x^2+