What is the maximum volume of a box that can be placed inside an ellipsoid $\frac{x^2}{16}+\frac{y^2}{9}+\frac{z^2}{25}=1$
The volume of a box is $V=xyz$ so I need to find $x,y,z$ with respect to the ellipsoid conditions and then find the maximum point.
$x=\sqrt{16-\frac{16y^2}{9}-\frac{16z^2}{25}}$
$y=\sqrt{9-\frac{9x^2}{16}-\frac{9z^2}{25}}$
$z=\sqrt{25-\frac{25x^2}{16}-\frac{25y^2}{9}}$
We take only the positive root as the length but on the other hand a negative root is just a length to the other direction and if 2 of the variables are negative the volume will be still a positive number
One simple standard procedure would be when you have the condition:
$$f(x,y,z)=\frac{x^2}{16}+\frac{y^2}{9}+\frac{z^2}{25}=1$$
and you want to maximize
$$V(x,y,z)=xyz$$
to set $$\vec\nabla f(x,y,z)=\lambda\cdot\vec\nabla V(x,y,z)$$
their gradient looking in the same direction. After that you have to choose the maxima out of the extrema.
This is the geometric approach learned in my analyses course.