How Do Lagrange Multipliers Optimize Ellipsoid Volume?

[Dethrone]

Use Lagrange multipliers to find $a,b,c$ so that the volume $V=\frac{4\pi}{3}abc$ of an ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$, passing through the point $(1,2,1)$ is as small as possible.

I just need to make sure my setup is correct.
$\triangledown f=(\frac{4\pi}{3}bc,\frac{4\pi}{3}ac,\frac{4\pi}{3}ab)$
$\triangledown g=(-2/a^3, -8/b^3, -2/c^3)$.
Where, $\triangledown f=\lambda \triangledown g$

 

[I like Serena]

Use Lagrange multipliers to find $a,b,c$ so that the volume $V=\frac{4\pi}{3}abc$ of an ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$, passing through the point $(1,2,1)$ is as small as possible.

I just need to make sure my setup is correct.
$\triangledown f=(\frac{4\pi}{3}bc,\frac{4\pi}{3}ac,\frac{4\pi}{3}ab)$
$\triangledown g=(-2/a^3, -8/b^3, -2/c^3)$.
Where, $\triangledown f=\lambda \triangledown g$

Hey Rido! ;)

Looks good to me! (Nod)