Find the absolute minimum of a function of $n$ variables

Question

$A$ is positive-defined matrix, and we define $f: \mathbb{R}^n \to \mathbb{R} $ by: $$f(x) = <Ax, x> + 2<b,x> +c$$ Where $b \in \mathbb{R}^n$ and $c \in \mathbb{R}$. Find the absolute minimum of the function. I was able to get the derivative, and I got $\nabla f(x) = 2Ax+2bx$ and therefore the derivative is $0$ for $x_0$ such that $Ax_0=-b$. For such $x_0$, we have $f(x_0)= <-b,x_0>+2<b,x_0>+c = <b,x_0>+c$

Now how do I prove this is the minimal value? I need to prove that $f(x) \geq <b,x_0>+c$
How do I do that?