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Math Help - Fermats criteria - Correct?

  1. #1
    Senior Member Twig's Avatar
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    Fermats criteria - Correct?

    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
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    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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