10. To inscribe a cylinder with the greatest total surface area in a cone with height $h$ and base radius $r$.
10. Denoting the radius of the base of the desired cylinder by \( x \), and its height by \( y \), we first find that \(\frac{x}{r} = \frac{h-y}{h}\), hence \( y = \frac{h(r-x)}{r} \).

Let the total surface area be \( z \), then \( z = 2 \pi x^{2} + 2 \pi x y \), or

\( z = \frac{2 \pi r x^{2} + 2 \pi h r x - 2 \pi h x^{2}}{r} \). We will investigate the function \( f(x) = -(h-r) x^{2} + h r x \). The maximum value is achieved at \( x = \frac{h r}{2(h-r)} \), and then we get \( y = \frac{h(h-2 r)}{2(h-r)} \); \( z = \frac{\pi r h^{2}}{2(h-r)} \); with the condition \( h > 2 r \).