Fermats criteria - Correct?

**Twig** · September 15th 2008, 06:26 AM

Fermats kriterium:

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

Fermat showed this in the 17th century.

THEOREM: $f: R \longrightarrow \ R$ , $x_0 \ \in \ D_f$

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

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

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

For $x > x_0$ :
$x - x_0 > 0$

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

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

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

DEFINITION: $f: R \longrightarrow \ R , x_0 \in \ R$

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

A STATIONARY POINT $x_0$ that is NOT an extremepoint is called a SADELPUNKT.

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

**Opalg** · September 15th 2008, 08:06 AM

In English, such a point is usually called a point of inflection (or you can spell it inflexion if you prefer).

The term saddle point is used for a point where a function of more than one variable has a stationary point that is not a local extremum. For a function of two variables, this usually means that some of the cross-sections through that point will have a local minimum there, and others will have a local maximum.

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