(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Proof: Sufficient condition for local extreme existence

Can you offer guidance or do you also need help?

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