# A rectangular box is to be inscribed inside the ellipsoid 2x^2 +y^2+4z^2 = 12. How do you find the largest possible volume for the box?

A rectangular box is to be inscribed inside the ellipsoid $2{x}^{2}+{y}^{2}+4{z}^{2}=12$. How do you find the largest possible volume for the box?
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The box volume is given by
$V=8|xyz|$
so the problem is:
Find $maxV\left(x,y,z\right)$ subjected to
$g\left(x,y,z\right)=2{x}^{2}+{y}^{2}+4{z}^{2}=12$
Using Lagrange multipliers we have the equivalent problem
Find the stationary points of
$L\left(x,y,z,\lambda \right)=V\left(x,y,z\right)+\lambda g\left(x,y,z\right)$
and verify the solutions which give a maximum for $V\left(x,y,z\right)$
The stationary ponts are obtained by solving for $x,y,z,\lambda$
$\nabla L\left(x,y,z,\lambda \right)=\stackrel{\to }{0}$ or
$\left\{\begin{array}{l}8yz-4\lambda x=0\\ 8xz-2\lambda y=0\\ 8xy-8\lambda z=0\\ 12-2{x}^{2}-{y}^{2}-4{z}^{2}=0\end{array}$
The solution is $\left(x=\sqrt{2},y=2,z=1\right)$ with corresponding volume
$V=16\sqrt{2}$