= Solution
For a regular smooth constraint $g=0$, meaning $\nabla g\ne0$ 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 $V=\pi r^2h$ and the surface-area constraint is $2\pi r^2+2\pi rh=A$. The multiplier equations are
$$
2\pi rh=\lambda(4\pi r+2\pi h),\qquad
\pi r^2=\lambda(2\pi r).
$$
The 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. \b[The largest volume is]
$$
\boxed{r=\sqrt{\frac A{6\pi}},\quad h=2r,\quad
V_{\max}=\frac{A^{3/2}}{3\sqrt{6\pi}}.}
$$
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