Proof: Sufficient condition for local extreme existence

[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.

 

[mighty2000]

The equality denoted by (*) is the Taylor expansion of a function at a point. It is a way of approximating a function using its derivatives at a specific point. In this case, we are using the Taylor expansion to approximate the function f(x) at a point x in a neighbourhood of a.

The Taylor expansion of a function f(x) at a point a is given by:

f(x) = f(a) + Df(a)(x-a) + \frac{1}{2}D^2f(a)(x-a, x-a) + ...

where Df(a) is the first derivative of f at a, D^2f(a) is the second derivative of f at a, and so on.

In the proof, we are using the Taylor expansion to approximate f(x) at a point x in a neighbourhood of a. This is why we have the term Df(a)(x-a) in the expansion.

The term D^2f(a\ +\ \gamma(x-a))(x-a, x-a) is the second derivative of f at a point a + \gamma(x-a), where \gamma is a small number between 0 and 1. This is because we are approximating f(x) at a point x in a neighbourhood of a, and we are using the second derivative at a point close to a (specifically, a + \gamma(x-a)).

So, the equality denoted by (*) is just the Taylor expansion of f(x) at a point x in a neighbourhood of a. It is a way of approximating f(x) using its derivatives at a specific point.

I hope this explanation helps!