Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 14 / a

Put $s=x^2+y^2$. The gradient components areA stationary point requires both components to vanish. If $x\ne 0$, then $s=1$; the second component subsequently forces $y=0$, giving $(\pm 1,0)$. If $x=0$, either $y=0$ or, writing $u=y^2$,Only $u=1/2$ is nonnegative. Thus there are five stationary points, with valuesThe cases $x=0$ and $s=1$ exhaust the first partial derivative equation, so no other stationary points have been lost by division.