不等式问题3设x,y,z为非负实
设x,y,z为非负实数。求证
(x^3+xyz)/(y+z)+(y^3+xyz)/(z+x)+(z^3+xyz)/(x+y)>=x^2+y^2+z^2 (1)
证明 (1)[(x^3+xyz)/(y+z)-x^2]+[(y^3+xyz)/(z+x)-y^2]+[(z^3+xyz)/(x+y)-z^2]>=0
[x/(y+z)](x-y)(x-z)+[y/(z+x)](y-z)(y-x)+[z/(x+y)](z-x)(z-y)>=0 (2)
由(2)的全对称性，不妨设x>=y>=z,则x/(y+z)>=y/(z+x),(x-y)(x-z)>=0,(z-x)(z-y)>=0,于是
...全部
设x,y,z为非负实数。求证
(x^3+xyz)/(y+z)+(y^3+xyz)/(z+x)+(z^3+xyz)/(x+y)>=x^2+y^2+z^2 (1)
证明 (1)[(x^3+xyz)/(y+z)-x^2]+[(y^3+xyz)/(z+x)-y^2]+[(z^3+xyz)/(x+y)-z^2]>=0
[x/(y+z)](x-y)(x-z)+[y/(z+x)](y-z)(y-x)+[z/(x+y)](z-x)(z-y)>=0 (2)
由(2)的全对称性，不妨设x>=y>=z,则x/(y+z)>=y/(z+x),(x-y)(x-z)>=0,(z-x)(z-y)>=0,于是
[x/(y+z)](x-y)(x-z)+[y/(z+x)](y-z)(y-x)+[z/(x+y)](z-x)(z-y)
>=[x/(y+z)](x-y)(x-z)+[y/(z+x)](y-z)(y-x)
>=[y/(z+x)](x-y)(x-z)+[y/(z+x)](y-z)(y-x)
=[y/(z+x)](x-y)[(x-z)-(y-z)]
=[y/(z+x)](x-y)^2
>=0,
因此，不等式(2)成立，从而不等式(1)成立。
。收起
