Let $f(x) = \langle Ax,x\rangle + 2\langle x,b\rangle +c$ where $A \in \mathbb R^{n \times n}, b \in \mathbb R^n, c \in \mathbb R$, $A$ is positive definite.
I want to show that $\lim_{|x| \to \infty} f(x) = \infty$
I know $\langle Ax,x\rangle$ is positive, but maybe $\langle x,b \rangle$ is negative so it evens out. It feels like there's some identity with inner products or formula that I'm missing, possible some trick with Cauchy-Schwarz.
HINT
Let $\vec x=\vec u t$ with $t\to \infty$ then
$$f(x) = t^2\langle Au,u\rangle + 2t\langle u,b\rangle +c$$