# Fermats criteria - Correct?

• Sep 15th 2008, 06:26 AM
Twig
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 $\displaystyle x=0$
• Sep 15th 2008, 08:06 AM
Opalg
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.