$11.(5$ e$.$c$.$ points$)$

Sometimes the solution of an optimization problem depends on the proportions of the shapes involved. As a case in point, suppose that a right circular cylinder of radius $\backslash ($ r $\backslash )$ and height $\backslash ($ h $\backslash )$ is inseribed in a right circular cone of radius $\backslash ($ R $\backslash )$ and height $\backslash ($ H $\backslash ),$ as shown above. Find the value of $\backslash ($ r $\backslash ),$ in terms of $\backslash ($ R $\backslash )$ and $\backslash ($ H $\backslash ),$ that maximizes the total surface area of the cylinder $($including the top and bottom.$)$ As you will see, the solution depends on whether $\backslash ($ H $\backslash$leqslant $2$ R $\backslash )$ and $\backslash ($ H$>2$ R $\backslash ).($SHOW ALL WORK$)$