Proposition 7

Let $f$ be a finite real function, defined on an interval $I \subset \mathbf{R}$. If $f$ admits a relative maximum (resp. relative minimum) at a point $x_0$ interior to I, and has both right and left derivatives at this point, then one has $f'_d(x_0) \leq 0$ and $f'_g(x_0) \geq 0$ (resp. $f'_d(x_0) \geq 0$ and $f'_g(x_0) \leq 0$); in particular, if $f$ is differentiable at the point $x_0$, then $f'(x_0) = 0$.

The proposition follows trivially from the definitions.

We can say further that if at a point $x_0$ interior to I the function $f$ is differentiable and admits a relative maximum or minimum, then the tangent to its graph is parallel to the abscissa. The converse is incorrect, as is shown by the example of the function $x^3$ which has zero derivative at the point $x = 0$, but has neither relative maximum nor minimum at this point.