Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 355 / 2 / a / Solution

For a cylinder of radius $r$ and length $L$, the mean curvature is $H=1/(2r)$ and the area is $2\pi rL$. Its Helfrich membrane energy per unit length is thereforeThe first term favors a narrow tube and the second penalizes its curvature. Setting the derivative with respect to $r$ to zero givesThe positive second derivative confirms a minimum.