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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?

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    $\begingroup$ 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 $p'(x)$. $\endgroup$ – Jyrki Lahtonen Nov 23 '16 at 17:39
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    $\begingroup$ It can still be injective if $p'$ has exactly one real zero (e.g. $p(x)= x^3$) $\endgroup$ – john Nov 23 '16 at 17:41
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    $\begingroup$ @JyrkiLahtonen , Ha thanks. I totally forgot that calculus is a thing. $\endgroup$ – Mike Pierce Nov 23 '16 at 17:55
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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.

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