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

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

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)

Resolved

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