Question

1.已知a大于b大于c,求证:a^2b+b^2c+c^2a大于ab^2+bc^2+ca^2 2.比较a^4-b^4与4a^3(a-b)

Answer 1

1)A>B>C
设A=B+X, C=B-Y,则有X,Y>0
因为A^2B+B^2C+C^2A-AB^2-BC^2-CA^2
=AB(A-B)+BC(B-C)+CA(C-A)
=ABX+BCY+CA(-X-Y)=AX(B-C)+CY(B-A)=AXY-CYX=XY(A-C)=XY(X+Y)
因为X>0,Y>0有XY(X+Y)>0
所以A^2B+B^2C+C^2A-AB^2-BC^2-CA^2>0
A^2B+B^2C+C^2A>AB^2+BC^2+CA^2
2)设B=A+X,则A-B=-X
则有 A^4-B^4=A^4-(A+X)^4=A^4-(A^4+4A^3X+6A^2X^2+4AX^3+X^4)
=-4A^3X-6A^2X^2-4AX^3-X^4
4A^3(A-B)=-4A^3X
则A^4-B^4-4A^3(A-B)=-6A^2X^2-4AX^3-X^4<0
所以A^4-B^4<4A^3(A-B)