limits - Prove that a second degree polynomial always has an extremum

This is one of those questions that I know intuitively, but find it hard to prove mathematically.

Problem

The idea is to try and prove that a second degree polynomial function, $f(x) = ax^2 + bx + c$, always has an extremum.

My thoughts

My thought is to try an explain that given $a > 0$, then $\lim\limits_{x\to\pm\infty}f(x) \to \infty$, so if a non-infinite value exists, it must have a bottom point between the extremes.

Likewise for $a<0$ but with a top point between two $-\infty$ extremes.

But I can't see how to prove this rigorously.