Proof of convexity of a quadratic function

Question

I have the next problem:

If $f(x)$ is a quadratic function with n variables:

$f(x) = 0.5$$\mathbf{x}^T$$A$$\mathbf{x}$$+$$\mathbf{b}$$^T$$\mathbf{x}$$+$$\mathbf{c}$

were $A$ is a symmetric matrix of $n$ x $n$. Proof that if $f$ is convex in $||x||<\beta$, then $f$ is convex in $\mathbb{R}^n$. I don't know how to start...

Thank you so much.

Answer 1

Thank you very much for the answer Autolatry but actually I am trying to get, first, the proof of convexity of $f$ on $||x||<\beta$ and second, I would like to now how can I relate this with the convexity of $f$ on $\mathbb{R}^n$. I would like to get a proof for a generic quadratic function without using directly the Hessian. There must be a way but no idea how to start...Have you got any idea? Thank you so much