Fermats criteria - Correct?

Fermats kriterium:

If takes in a (local) extremvalue and can be differentiated in , then

Fermat showed this in the 17th century.

__THEOREM:__ ,

If takes in an extremevalue and can be differentiated in then

__PROOF:__ Look at

If is local maximum: for alla in a surrounding (close).

For :

and

For :

and

This about is such an important quality that it has its own name:

__DEFINITION__:

is called a STATIONARY POINT TO , if can be differentiated in and

A __STATIONARY POINT__ that is NOT an extremepoint is called a __SADELPUNKT__.

Example) has a SADELPUNKT (terasspunkt) in