A cylinder of greatest curved surface area is inscribed in a cone of base radius $r$. The radius of this cylinder is

1. With cone height $h$, cylinder height $=\dfrac{h(r-x)}{r}$ where $x$=cylinder radius. 2. Curved surface $S(x)=\dfrac{2\pi h}{r}(rx-x^2)$. 3. $S'(x)=\dfrac{2\pi h}{r}(r-2x)=0 \Rightarrow x=\dfrac{r}{2}$. 4. $S''(x)=\dfrac{-4\pi h}{r}<0$, so it is a maximum. _Source: NCERT Class 12 Mathematics Ch 6 "Application of Derivatives", p.23_