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How do we prove that among all closed surfaces with a given surface area, the sphere is the one that encloses the largest volume, and not do it by cases?

so far I've tried is that I know the formula for the surface of the sphere and volume of sphere

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    http://en.wikipedia.org/wiki/Isoperimetric_inequality2011-07-19
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    You certainly cannot prove this result simply by examining a bunch of closed surfaces and comparing volume enclosed. There are infinitely many different closed surfaces, so you cannot test them all. The problem is not entirely trivial (compare with the 2-dimensional case, with closed simple curves, area, and length).2011-07-19
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    The is a really good elementary discussion of this problem at http://www.cut-the-knot.org/do_you_know/isoperimetric.shtml2011-07-20
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    Perhaps use the calculus of variations and the Euler-Lagrange formula.2012-11-14

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