Setting up and minimizing surface area of a cylinder

From the constraint $\pi r^2 h = 500$, solve for $h$: $h = \dfrac{500}{\pi r^2}$. Substituting into $S = 2\pi r^2 + 2\pi r h$ gives $S(r) = 2\pi r^2 + 2\pi r \cdot \dfrac{500}{\pi r^2} = 2\pi r^2 + \dfrac{1000}{r}$, valid for $r>0$.

Differentiate: $S'(r) = 4\pi r - \dfrac{1000}{r^2}$. Setting $S'(r)=0$ gives $4\pi r^3 = 1000$, so $r^3 = \dfrac{250}{\pi}$ and $r = \left(\dfrac{250}{\pi}\right)^{1/3}$. Since $S'(r) < 0$ for small $r$ and $S'(r) > 0$ for large $r$ (or checking $S''(r) = 4\pi + \frac{2000}{r^3} > 0$), this critical point is indeed a minimum.