Maximize the volume of cuboid with fixed surface $S > 0$.

Question

I'm trying to maximize the volume of a cuboid with surface $S > 0$.

I found this question under the topic of "extreme points of manifold with boundary", so now I asked myself if I should maximize the function $f(x,y,z) = xyz$, what my constraints are apart from $x,y,z \geq 0$? And is $g(x,y,z) = 2xy+2xz+2yz - S$ a constraint? Thanks for advice.

Answer 1

Yes, $g(x,y,z) = 0$ is a constraint to the function $f(x,y,z) = xyz$.