$$\begin{gathered} \frac{{\mathrm{h}}^2}{4}+{\mathrm{r}}^2=144 \\ \Rightarrow {\mathrm{r}}^2=144-\frac{{\mathrm{h}}^2}{4} \end{gathered}$$

Volume of the cylinder

$$\begin{gathered} \mathrm{V}={\mathrm{\pi r}}^2\mathrm{h} \\ =\mathrm{\pi }\left(144-\frac{{\mathrm{h}}^2}{4}\right)\mathrm{h} \\ =\mathrm{\pi }\left(144\mathrm{h}-\frac{{\mathrm{h}}^3}{4}\right) \\ {\mathrm{V}}^{\mathrm{I}}=\mathrm{\pi }\left(144-\frac{3{\mathrm{h}}^2}{4}\right) \end{gathered}$$

V is Min or Max

$$\begin{gathered} \Rightarrow {\mathrm{V}}^{\mathrm{I}}=0 \\ 144=\frac{3{\mathrm{h}}^2}{4} \\ \Rightarrow {\mathrm{h}}^2=\left(48\right)4=\left(64\right)3 \\ \mathrm{h}=8\sqrt{3} \end{gathered}$$