Wikipedia:Reference desk/Archives/Mathematics/2011 February 26

= February 26 =

Lagrangian Multipliers
I'm having some difficulties with using Lagrangian multipliers correctly, and would appreciate any help anyone can give. I thought I'd start with a simple problem that I can easily check the answer to, but I can't get the right answer. In particular, what is the largest volume of cube that will fit inside a unit sphere? My attempted solution is below.


 * $$f = xyz$$
 * $$g = x^2 + y^2 + z^2 - 1$$
 * $$L = f - \lambda gx = xyz - \lambda (x^2 + y^2 + z^2 - 1)$$

Hence:
 * $$L_x = yz - 2\lambda x = 0$$
 * $$L_y = xz - 2\lambda y = 0$$
 * $$L_z = xy - 2\lambda z = 0$$
 * $$L_\lambda = -(x^2 + y^2 + z^2 - 1) = 0$$

Using Wolfram Alpha to solve the simultaneous equations, to prevent me from messing that bit up, I get the following (ignoring non-physical negative solutions): :$$L = \tfrac{1}{2\sqrt{3}}, x = \tfrac{1}{\sqrt{3}}, y = \tfrac{1}{\sqrt{3}}, z = \tfrac{1}{\sqrt{3}}$$ This is obviously not correct. Where have I gone wrong? --131.111.184.8 (talk) 21:31, 26 February 2011 (UTC) Please ignore the above, I've just realised what I've done... (for the interested, side length is twice x...) --131.111.184.8 (talk) 21:35, 26 February 2011 (UTC)

StuRat (talk) 00:55, 1 March 2011 (UTC)