Question

find the volume of largest rectangular box with faces parllel to coordinate planes that can be inscribed in the ellipsoid 16x^2 +4y^2+9z^2=144

Answer 1

You could use a lagrange multiplier
the ellipsoid is symmetric about its center at (0,0,0), and consequently sides of the box
the constraint is that the x,y,z values must satisfy (line on) the ellipsoid:
\[ V= 8xyz + \lambda(16x^2 +4y^2+9z^2-144 ) \]

Can you try to solve this? (It's a bit painful, but doable)