# Proof: Sufficient condition for local extreme existence

1. Jan 26, 2006

### twoflower

Let

$$G \subset \mathbb{R}^{n}\mbox{ open }$$

$$a \in G$$

$$f : G \rightarrow \mathbb{R}$$

$$f \in C^{1}(G)$$

$$Df(a) = \overrightarrow{0}$$

Then:

(i) if $D^2f(a)$ is positively definite, then f has local minimum in $a$

(ii) if $D^2f(a)$ is negatively definite, then f has local maximum in $a$

(iii) if $D^2f(a)$ is indefinite, then f doesn't have extreme in $a$

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.

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

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

Then

$$\forall x \mbox{ in this neighbourhood } \exists\ \gamma \in (0,1) \mbox{ such that }$$

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

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

I don't get the equality denoted by $(*)$.

Could someone explain this to me?

Thank you very much.

Last edited: Jan 26, 2006