= Minimum-area closed cylinder at fixed volume
{title2=$h=2r,\quad A_{\min}=3(2\pi)^{1/3}V^{2/3}$}
For radius $r>0$ and height $h>0$, volume is $\pi r^2h$ and area including both ends is $2\pi r^2+2\pi rh$. The <Lagrange multiplier> equations give $h=2r$. Substituting the constraint gives the one-variable function $2\pi r^2+2V/r$, which tends to infinity at both ends and has its only stationary point at $r^3=V/(2\pi)$. This proves the stationary cylinder is the unique global minimizer.
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