4.11 Discriminant test

The following describe all possible non-degenerate stationary points $P=(a,b)$ of a function $f$ of two variables:

(i) if $(f_{xx})_{P}>0$ and $\Delta_{P}>0$, then $f$ has a local minimum at $P$;

(ii) if $(f_{xx})_{P}<0$ and $\Delta_{P}>0$, then $f$ has a local maximum at $P$;

(iii) if $\Delta_{P}<0$, then $P$ is a saddle.