minimum surface are for a given volume

Question

How to prove that for a given enclosed volume, a sphere has minimum surface area

Answer 1

I think the answer to your question is "you can't".

You've stumbled on a hard problem - well beyond 12th grade calculus: see Prove that the sphere is the only closed surface in $\mathbb{R}^3$ that minimizes the surface area to volume ratio. .

The two dimensional version - the circle minimizes perimeter for a fixed area - might be accessible. Search for isoperimetric problem for many good links.