How can you quickly tell if a cubic polynomial gives an injective function?

Question

Given a cubic polynomial $p = ax^3+bx^2+cx+d$ with real coefficients, is there a quick way to determine if the function $p \colon \mathbb{R} \to \mathbb{R}$ is injective? Does anyone know if there is a clean classification of cubic polynomials that induce injective functions?

Answer 1

Since you asked for a quick way, certainly the discriminant of the first derivative is the way to go, but that can be boiled down a bit further.

From $3ax^2+2bx+c=0$ we obtain the requirement $D=4(b^2-3ac)\le0$ which reduces to the quick test

$$ b^2\le 3ac $$

for $p$ to be injective.