求函数最小值
设x,y,z>0,且x+y+z=6.求函数
f(x,y,z)=(x^2+y^2+z^2)/[(yz)^2+(zx)^2+(xy)^2]
的最小值.
解 函数f(x,y,z)当x=0,y=z=3时,有最小值.最小值为2/9.
即证
(x^2+y^2+z^2)/[(yz)^2+(zx)^2+(xy)^2]>=2/9 (1)
即证
(x^2+y^2+z^2)*(x+y+z)^2/[(yz)^2+(zx)^2+(xy)^2]>=8 (2)
(2)<===>
Σx^4+2Σ(y+z)x^3-6Σ(yz)^2+2xyzΣx>=0 (3)
设x=min(x,y,z),(3)分解为
x(x+3y+3z)*(x-y)*(x-z)+xyz(x+y+z)
+[-3x^2+2x(y+z)+y^2+z^2+4yz]*(y-z)^2>=0
显然成立.