Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 4 / c / Solution

The gradient vanishes when $4x+3y-1=0$ and $3x+2y-1=0$. Subtracting these equations gives $x+y=0$, so the unique critical point is $\boxed{(x,y)=(1,-1)}$, with $f(1,-1)=0$.

The Hessian matrix isIt is indefinite, so the critical point is a saddle point. Explicitly, around that point the change is exactly $2u^2+3uv+v^2$: it is positive for $v=0,u\ne 0$ and negative for $v=-3u/2,u\ne 0$. Thus arbitrarily close points have both larger and smaller values.