- #1
twoflower
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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.
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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