Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 5 / b / Solution

The Hessian matrix isIts eigenvalues have opposite signs, so the stationary point is a saddle point. More directly, putting $x=1+u$ and $y=-1+v$ gives $f=u^2+uv-v^2$: it increases along $v=0$ and decreases along $u=0$.