Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 4A (Variational Principles)

Introduce the orthogonal coordinatesThen the constraint and objective becomeThe Lagrange multiplier equations for a stationary point areIf $\lambda =0$, then $u=y=0$ and $v=\pm 1$, givingIf $\lambda \ne 0$, then $v=0$. The first two equations and the constraint implyConverting back to $(x,y,z)$ gives the other four points. Therefore all stationary points arewith matching signs, together withThe middle pair has $\varphi =1/2$, the last pair has $\varphi =-1/2$, and the first pair has $\varphi =0$.