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Mathematics-Online course: Prepcourse Mathematics - Analysis - Extrema and Curve Sketching

Test for Extrema

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A $ n$ -times continuously differentiable function $ f$ with

$\displaystyle f'(a) =f''(a)=\cdots=f^{(n-1)}(a)=0$   and$\displaystyle \quad f^{(n)}(a) \neq 0\,, $

has an extremum at $ a$ iff $ n$ is even. In this case $ f$ has a local maximum (minimum) at $ a$ , if $ f^{(n)}(a)<0$ ( $ f^{(n)}(a)>0$ ).
(Authors: App/Höllig/Abele)
The polynomial

$\displaystyle p(x) = (x+1)^2 x^3 (x-1)^4 $

has a double zero point at $ x=-1$, a triple zero point at $ x=0$ and a quadruple zero point at $ x=1$.
\includegraphics[width=8.4cm]{Extremwerttest.eps}

According to Leibnitz's Rule it is

$\displaystyle p(-1) = p'(-1) = 0, \quad p''(-1) = 2(-1)^3(-2)^4 < 0\,. $

So, at $ x=-1$ $ p$ has a maximum. At $ x=1$ $ p$ has a minimum, since

$\displaystyle p(1) = \hdots = p'''(1) = 0, \quad p^{(4)}(1) = 2^2 1^3 4! > 0\,. $

Finally $ x=0$ is no extremum of $ p$, for

$\displaystyle p(0) = p'(0) = p''(0) = 0, \quad p'''(0) = 3! (-1)^4 \ne 0 $

and the algebraic sign changes.
(Authors: App/Höllig/Abele )
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  automatically generated 9/18/2007