Timeline for Injectivity of $f_r : \mathbb{R} \to \mathbb{R}$ Defined as $f_r(x) = x^3 + rx + 1$: Inflection Points and Injectivity

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Mar 29, 2018 at 13:33	comment	added	The Pointer		I mean, if $r = 0$, then the derivative $f' > 0$ at all points except at $x = 0$, since it is an inflection point. So, generally speaking, if a function has a strictly positive or negative derivative at all points except at the inflection points, must this necessarily mean that the inflection points don't "break" injectivity? For instance, we could imagine a function with $f' > 0$ everywhere except at multiple inflection points where $f' = 0$ and $f'' = 0$.
Mar 29, 2018 at 13:28	comment	added	Mundron Schmidt		I don't understand your question because I talked about a family of functions, where the inflection point marks the boundary of the injective functions. But you talk about one injective function on some interval.One specific injective function doesn't need to have an inflection point. And you can also have a family of injective functions without inflection points.
Mar 29, 2018 at 13:20	comment	added	The Pointer		Thanks for the response. So, if a function is injective along some interval, must that interval necessarily also include the inflection point? After all, as you say, the inflection point is simply a boundary.
Mar 29, 2018 at 13:17	history	answered	Mundron Schmidt	CC BY-SA 3.0