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

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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.

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Ethan Bolker
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  • @ResearchEngineer I suspect you studied the problem in two dimensions (the isoperimetric problem). That's accessible, though not plain calculus. – Ethan Bolker Apr 07 '17 at 18:54