Vanishing of the Derivative at an Interior Local Extremum

Let $\mathbb{R}$ be the set of real numbers, with its order $\le$ and the associated strict order $<$, regarded as a metric space through the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line.

Let $p,q\in\mathbb{R}$, let $g:(p,q)\to\mathbb{R}$ be a function on the open interval $(p,q)$, and let $c\in(p,q)$ be a point at which $g$ is differentiable.

If $g$ has a local maximum at $c$ relative to $(p,q)$, or a local minimum at $c$ relative to $(p,q)$, then