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Thread: Fermats criteria - Correct?

  1. #1
    Senior Member Twig's Avatar
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    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 $\displaystyle x=0$
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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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