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

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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): :$L=\frac{1}{2\sqrt{3}},x=\frac{1}{\sqrt{3}},y=\frac{1}{\sqrt{3}},z=\frac{1}{\sqrt{3}}$ 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)

Template:ResolvedStuRat (talk) 00:55, 1 March 2011 (UTC)