Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 19 / a / Solution

For a closed circular cylinder, $S=2\pi r^2+2\pi rh$ and the volume constraint is $\pi r^{2}h=V$. Use a Lagrange multiplier:The stationary equations areFor $r>0$, the second gives $\lambda =2/r$; the first then gives $h=2r$. The constraint yieldsandTo verify a global minimum, eliminate $h$ to obtain $S(r)=2\pi r^2+2V/r$. It tends to infinity at both ends of $r>0$ and has $S^{\prime \prime }(r)=4\pi +4V/r^3>0$, so its unique stationary point is the global minimum.