大师作文网已知x＞y,求证x³-y³＞2x²y-2xy²
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已知x＞y,求证x³-y³＞2x²y-2xy²
x³-y³-(2x²y-2xy²)=x³-y³-2xy(x-y)=(x-y)(x^2+xy+y^2)-2xy(x-y)
=(x-y)(x^2-xy+y^2)
x>y>=0,(x-y)(x^2-xy+y^2)=(x-y)[x(x-y)+y^2)]>0
x>=0>y,(x-y)(x^2-xy+y^2)=(x-y)[x^2+x(-y)+y^2)]>0
0>=x>y,(x-y)(x^2-xy+y^2)=(x-y)[(x-y)^2+xy]>0
x＞y,所以x³-y³＞2x²y-2xy²
=(x-y)(x^2-xy+y^2)
x>y>=0,(x-y)(x^2-xy+y^2)=(x-y)[x(x-y)+y^2)]>0
x>=0>y,(x-y)(x^2-xy+y^2)=(x-y)[x^2+x(-y)+y^2)]>0
0>=x>y,(x-y)(x^2-xy+y^2)=(x-y)[(x-y)^2+xy]>0
x＞y,所以x³-y³＞2x²y-2xy²
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