Question

I need help with local extremes. I'm writing problem in comment

Answer 1

F(x)=<Ax,x>+<2b,x>+c
A is real, regular, symmetric, positive-definite matrix; x, a, b are vectors from R^n, c is real number.

I need to determine necessary condition for local extreme.
I know that F<x+h>-F<x> must be 0.

F<x+h>-F<x>
=<Ax,x>+<Ax,h>+<Ah,x>+<Ah,h>+<2b,x>+<2b,h>+c-<Ax,x>-<2b,x>-c
=<Ax,h>+<Ah,x>+<Ah,h>+<2b,h>
=2<Ax,h>+<2b,h>=2<Ax+b,h>

Why <Ax,h>=<Ah,x> and why <Ah,h>=0?
How to find F'(x) and F''(x)?

Answer 2

use the definition of differentiation or Taylor series or Big O, simple straight forward

Answer 3

can you write it?

Answer 4

I have written that F'(x)=2(Ax+b) and F''(x)=2A, but I don't know how did they get it. And if <Ah,h> goes to 0 because h goes to 0, why other scalar products are different from 0?