A right solid circular cylinder of given volume will have the least total surface area when

Let r be the radius, h be the height, V be the volume and S be the total surface area of a cylinder.

$V=\pi r^{2}h\Rightarrow h=\frac{V}{\pi r^2}.$ Now $S=2\pi rh+2\pi r^2=2\pi r\frac{V}{\pi r^2}+2\pi r^2=\frac{2V}{r}+2\pi r^2$

$\frac{dS}{dr}=\frac{-2V}{r^2}+4\pi r\Rightarrow \frac{d^{2}S}{d{r}^2}=\frac{4V}{r^3}+4\pi >0.$

$\frac{dS}{dr}=0\Rightarrow \frac{-2V}{r^2}+4\pi r=0\Rightarrow \frac{2V}{r^2}=4\pi r\Rightarrow V=2\pi r^3\Rightarrow \pi r^{2}h=2\pi r^3\Rightarrow h=2r$

$\therefore S$ has minimum when $h=2r.$