已知：xy≠0且x≠y比較M=x^4-y^4與N=4x^3(x-y)的大小

熱心網(wǎng)友

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)^20,[(X+Y)^2+2X^2]0∴－(X－Y)^2[(X+Y)^2+2X^2]<0即: M

熱心網(wǎng)友

N=4X^4-4YX^3M=X^4-Y^4所以N大于M

熱心網(wǎng)友

M大

熱心網(wǎng)友

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 假設(shè)原式要等于0,只有 2x^2+(x+y)^2=0此時(shí) -y = x = 0 , 即 x=y=0與x≠y不合。。。所以 2x^2 + (x+y)^2 ≠ 0 且大于0所以 M-N < 0即 M < N(說(shuō)明一下。。。由于前次回答,是我看見(jiàn)有一樣的題目,而且是被采納了的。。所以直接復(fù)制了一下,sorry,所以沒(méi)好好看題。。。現(xiàn)在修改過(guò)了。。。一定正確)。