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March 21st, 2017, 06:54 PM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3  Partial Derivative Application for 3D object
Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid 2x^2 + 72y^2 + 18z^2 = 288. Answer: 256/sqrt(3) I would be much appreciated if anyone kindly point me to the right direction. Thanks! 
March 21st, 2017, 07:33 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,501 Thanks: 1372 
well by symmetry your corner points will be $(\pm a, \pm b, \pm c)$ with volume $V=8abc$ you can reduce this a bit using the formula for the ellipsoid to get points at $\left(\pm a, \pm b, \pm \sqrt{\dfrac{288  2a^2  72b^2}{18}}\right)$ with volume $V=8ab \sqrt{\dfrac{288  2a^2  72b^2}{18}}$ As usual take the gradient of this expression, with respect to $a,~b$ and set it equal to the vector 0 and solve for $a,~b$ Solving this might be a bit of a mess but Mathematica returns a pretty digestible answer so it might not be that bad. The way to ensure that this extreme points is a maximum is a bit more complicated for a 3D than a 2D case. Do you know how to use the two dimensional 2nd derivative test to do this? Last edited by romsek; March 21st, 2017 at 08:13 PM. 

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