You construct a rectangular Box with volume K cm^3 Prove that a cube uses the least amount of material to construct the box

Hint: If you know calculus and are allowed to use Lagrange multipliers, use them to find a global maximum for the volume $abc$ assuming the amount of material for the box $2(ab + bc + ca)$ is fixed, and $a,b,c > 0$. It's pretty obvious that a global maximum must exist which is a local maximum for $a,b,c > 0$, and no  local minimum exists for positive $a,b,c$ because you could always make one side arbitrarily small and take the volume to zero.
