4.9 Classification of stationary points via the discriminant

The three main types of stationary points are:

(i) local minima – which look like bowls;

(ii) local maxima – which look like hill tops;

(iii) saddles – which are also known as mountain passes or cols.

See graphs on subsequent pages. To distinguish between them we use the Hessian discriminant

We write $\Delta_{P}$ for the discriminant evaluated at $P$. The stationary point $P$ is non-degenerate if $\Delta_{P}\neq 0$. Take care with the sign and the square here.