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Minimizing distance from the origin

Minimize x^2 + y^2 + z^2 subject to 3x^2 + (root 3)y^2 - (root 3)z^2 = 1.

I can see its at a minimum when (x,y,z)=(1/(root3),0,0) by thinking about the shape of the surface 3x^2 + (root 3)y^2 - (root 3)z^2 = 1, but I can't see how to actually deduce this. Any ideas?

Reply 1

steve10

You do this kind of problem using a Lagrange multiplier.

You have f(x) = x^2 + y^2 + z^2
and
g(x) = 3x^2 + (root 3)y^2 - (root 3)z^2 - 1. (Note that g(x) = 0)

So, now you want to minimise F(x,s) = f(x) - s*g(x) (usually lambda is used instead of s)

So take all the partial derivatives for F wrt x,y,z and s, and set them equal to zero. You will now have a set of 4 simultaneous eqns to solve for x,y,z and s. For x,y and z you should get (1/(root3),0,0) <-- using what you wrote down.