Past exam of the mathematical methods II course of the University of Oxford / 2014 resit / 6 / b / Solution

The gradient is $(2x+2,-2y-4)$, so the only stationary point is $(x,y)=(-1,-2)$. Its Hessian matrix is $\mathrm{diag}(2,-2)$ and its determinant is $-4<0$. ThusCompleting squares confirms the classification directly: $f=(x+1{)}^2-(y+2{)}^2+13$. Arbitrarily close to the point, moving in the $x$ direction increases the value and moving in the $y$ direction decreases it.