 Solution

For a regular smooth constraint $g=0$, meaning $\nabla g\ne 0$ at the point, constrained stationarity requires $\nabla f=\lambda \nabla g$. Solve these Lagrange multiplier equations together with the constraint, then compare candidate values and possible boundary or singular cases.

Let $r>0$ and $h>0$ be cylinder radius and height. Its volume is $V=\pi r^{2}h$ and the surface-area constraint is $2\pi r^2+2\pi rh=A$. The multiplier equations areThe second gives $\lambda =r/2$, and substituting into the first gives $h=2r$. Consequently $6\pi r^2=A$. To check global maximality, eliminate $h$ and obtain $V(r)=Ar/2-\pi r^3$ on $0<r<\sqrt{A/(2\pi )}$; it tends to zero at both ends and has exactly this one stationary point. The largest volume is