数学问题，请教！！

This is a question about Lagrange multipliers
Let f(x,y,z)=100-x^2-y^2-z^2; g(x,y,z)=x+2y+z-a
To find the optimal value of x,y,z that maximize f(x,y,z), set





圖片參考：http://mathworld.wolfram.com/images/equations/LagrangeMultiplier/equation2.gif



for all
圖片參考：http://mathworld.wolfram.com/images/equations/LagrangeMultiplier/inline23.gif
is called the Lagrange multiplier.
We have∂f ∂ x
-2x+λ=0
-2y+2λ=0
-2z+λ=0
So λ=2x=y=2z
x=λ/2,y=λ,z=λ/2
substitute into x+2y+z=a
a=3λ,λ=a/3
So x=a/6,y=a/3,z=a/6
The optimal value function v(a)
=100-x^2-y^2-z^2
=100-(a/6)^2-(a/3)^2-(a/6)^2
=100-a^2/6
For example when a=30
x=5,y=10,z=5
v(30)=-50


2007-10-25 05:12:24 補充：
∂f ∂ x 多餘了﹐我原本預先copy時會用到的