Fermats criteria - Correct?

Fermats kriterium:

If $\displaystyle f $ takes in $\displaystyle x_0$ a (local) extremvalue and can be differentiated in $\displaystyle x_0$, then $\displaystyle f'(x_0) \ = 0 $

Fermat showed this in the 17th century.

__THEOREM:__ $\displaystyle f: R \longrightarrow \ R $ , $\displaystyle x_0 \ \in \ D_f $

If $\displaystyle f $ takes in $\displaystyle x_0$ an extremevalue and $\displaystyle f$ can be differentiated in $\displaystyle x_0$ then $\displaystyle f'(x_0) \ = \ 0 $

__PROOF:__ Look at $\displaystyle \frac{\triangle f}{\triangle x} \ = \ \frac{f(x) - f(x_0)}{x - x_0}$

If $\displaystyle x_0$ is local maximum: $\displaystyle f(x) \leq f(x_0) $ for alla $\displaystyle x $ in a surrounding (close).

For $\displaystyle x > x_0$ :

$\displaystyle x - x_0 > 0 $

and $\displaystyle \frac{\triangle f}{\triangle x} \ = \ \frac{f(x) - f(x_0)}{x - x_0}$ $\displaystyle \leq 0 \ \Rightarrow \ \lim_{\triangle x \rightarrow 0+} \frac{\triangle f}{\triangle x} = f'(x_0) \ \leq \ 0 $

For $\displaystyle x < x_0$:

$\displaystyle x - x_0 < 0 $ and $\displaystyle \frac{\triangle f}{\triangle x} \ = \ \frac{f(x) - f(x_0)}{x - x_0}$ $\displaystyle \geq 0 \Rightarrow \lim_{\triangle x \rightarrow 0-} \frac{\triangle f}{\triangle x} = f'(x_0) \geq 0 $

This about $\displaystyle f'(x_0) = 0 $ is such an important quality that it has its own name:

__DEFINITION__: $\displaystyle f: R \longrightarrow \ R , x_0 \in \ R $

$\displaystyle x_0 $ is called a STATIONARY POINT TO $\displaystyle f $, if $\displaystyle f $ can be differentiated in $\displaystyle x_0$ and $\displaystyle f'(x_0) = 0$

A __STATIONARY POINT__ $\displaystyle x_0$ that is NOT an extremepoint is called a __SADELPUNKT__.

Example) $\displaystyle f(x) = x^3 $ has a SADELPUNKT (terasspunkt) in $\displaystyle x=0$