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

For a differentiable function at an interior stationary point, both first partial derivatives vanish. If the second derivatives are continuous, form the symmetric Hessian matrixThe second-derivative test gives a strict local minimum if $D>0$ and $f_{xx}>0$, a strict local maximum if $D>0$ and $f_{xx}<0$, and a saddle point if $D<0$. If $D=0$, higher-order terms must be examined. For example, $x^4+y^4$, $-x^4-y^4$ and $x^4-y^4$ all have zero Hessian matrix at the origin but respectively a minimum, maximum and saddle. These are local interior criteria; boundary extrema require separate treatment.

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.