已知：xy≠0且x≠y
比较M=x^4-y^4与N=4x^3(x-y)的大小
M－N
=X^4－Y^4－4X^3(X－Y)
=(X－Y)(－3X^3+X^2Y+XY^2+Y^3)
=(X－Y)[(－2X^3+X^2Y+XY^2)－X^3+Y^3]
=(X－Y)[－X(2X^2－XY－Y^2)－(X－Y)(X^2+XY+Y^2)]
=－(X－Y)^2(3X^2+2XY+Y^2)
=－(X－Y)^2[(X+Y)^2+2X^2]
因XY≠0,∴X≠0,Y≠0,
又X≠Y
∴(X－Y)^2>0,[(X+Y)^2+2X^2]>0
∴－(X－Y)^2[(X+Y)^2+2X^2]<0
即: M<N
M-N=(x^2+y^2)(x+y)(x-y)-4x^3(x-y)
=(x-y)(x^3+y^3+yx^2+xy^2-4x^3)
=-(x-y)(3x^3-y^3-xy^2-yx^2)
=-(x-y)(2x^3-2yx^2+x^3-y^3+yx^2-xy^2)
=-(x-y)[2x^2(x-y)+(x-y)(x^2+xy+y^2)+xy(x-y)]
=-(x-y)(x-y)(2x^2+x^2+xy+y^2+xy)
=-(x-y)^2[2x^2+(x+y)^2]
=-(x-y)^2[2x^2+(x+y)^2]
由于x≠y，所以 x-y≠0
而 假设原式要等于0,只有 2x^2+(x+y)^2=0
此时 -y = x = 0 , 即 x=y=0
与x≠y不合...所以 2x^2 + (x+y)^2 ≠ 0 且大于0
所以 M-N < 0
即 M < N
(说明一下...由于前次回答,是我看见有一样的题目,而且是被采纳了的..所以直接复制了一下,sorry,所以没好好看题...现在修改过了...一定正确)
M大
N=4X^4-4YX^3
M=X^4-Y^4
所以N大于M
