代数不等式（2）设x,y,z为正实数,且x+y+z=3.求证 2(x^3+y^3+z^3)+9>=5(x^2+y^2+z

2(x^3+y^3+z^3)+9
=2(x3+y3+z3)+(x+y+z)^2
=2(x3+y3+z3+xy+xz+yz)+x2+y2+z2
化简,知原不等式可化为
x3+y3+z3+xy+xz+yz>=2x2+2y2+2z2
由著名的分解
x3+y3+z3=3xyz+(x+y+z)(x2+y2+z2-xy-yz-xz)
而x+y+z=3,知欲证不等式可化为
3xyz+x2+y2+z2-2xy-2xz-2yz>=0
或3xyz+(x-y)^2+(y-z)^2+(x-z)^2-x2-y2-z2>=0
此不等式显然,命题得证.

2(x^3+y^3+z^3)+9>=5(x^2+y^2+z^2)