Minimum-area closed cylinder at fixed volume ($h=2r,A_{\min }=3(2\pi {)}^{1/3}{V}^{2/3}$)

For radius $r>0$ and height $h>0$, volume is $\pi r^{2}h$ 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.