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Proof: Sufficient condition for local extreme existence

  1. Jan 26, 2006 #1
    Let


    [tex]
    G \subset \mathbb{R}^{n}\mbox{ open }
    [/tex]

    [tex]
    a \in G
    [/tex]

    [tex]
    f : G \rightarrow \mathbb{R}
    [/tex]

    [tex]
    f \in C^{1}(G)
    [/tex]

    [tex]
    Df(a) = \overrightarrow{0}
    [/tex]

    Then:

    (i) if [itex]D^2f(a)[/itex] is positively definite, then f has local minimum in [itex]a[/itex]

    (ii) if [itex]D^2f(a)[/itex] is negatively definite, then f has local maximum in [itex]a[/itex]

    (iii) if [itex]D^2f(a)[/itex] is indefinite, then f doesn't have extreme in [itex]a[/itex]


    Unfortunately we weren't given an entire proof at the lecture, but here's what we had been told:

    Hint of the proof:

    (i) First step: Won't tell you neither examinate it.

    [tex]
    D^2f(a)\mbox{ P.D } \Rightarrow \exists \mbox{ neighbourhood of } a \mbox{ on which it is P.D} \Rightarrow
    [/tex]

    [tex]
    \exists\ \xi > 0:\ \ Df(x)(h,h) \geq \xi \parallel h \parallel^2 \forall x \mbox{ from this neighbourhood }
    [/tex]

    Then

    [tex]
    \forall x \mbox{ in this neighbourhood } \exists\ \gamma \in (0,1) \mbox{ such that }
    [/tex]

    [tex]
    \mbox{(*) } f(x)\ -\ f(a)\ -\ Df(a)(x-a) = \frac{1}{2}D^2f(a\ +\ \gamma(x-a))(x-a, x-a) \Rightarrow
    [/tex]

    [tex]
    f(x) = f(a)\ +\ \frac{1}{2}D^2f(a\ +\ \gamma(x-a))(x-a, x-a) > f(a) \Rightarrow \mbox{ local minimum in } a
    [/tex]


    I don't get the equality denoted by [itex](*)[/itex].

    Could someone explain this to me?

    Thank you very much.
     
    Last edited: Jan 26, 2006
  2. jcsd
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