Classification of Stationary Points

Theorem

Function of One Variable

Let $f$ be a real function which is differentiable on the open interval $\openint a b$.

Let $P$ be a stationary point of $f$.

Then $P$ is either:

a local maximum

a local minimum

a point of inflection.

Function of Two Variables

Let $\SS$ be a surface defined by the Cartesian equation $z = \map f {x, y}$.

Let $P$ be a stationary point on $\SS$.

Then $P$ is either:

a local maximum

a local minimum

a saddle point.