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

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?

• It is injective iff it is everywhere increasing or decreasing iff $p'(x)$ has no real zeros. You only need to calculate the discriminant of the quadratic polynomial $p'(x)$. – Jyrki Lahtonen Nov 23 '16 at 17:39
• It can still be injective if $p'$ has exactly one real zero (e.g. $p(x)= x^3$) – john Nov 23 '16 at 17:41
• @JyrkiLahtonen , Ha thanks. I totally forgot that calculus is a thing. – Mike Pierce Nov 23 '16 at 17:55

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.