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

10 events

when toggle format	what		by	license	comment
Mar 29, 2018 at 17:47	history	edited	Robert Z	CC BY-SA 3.0	added 33 characters in body
Mar 29, 2018 at 13:47	vote	accept	The Pointer
Mar 29, 2018 at 13:32	comment	added	Robert Z		No, $x^2$ is strictly increasing in $(0,+\infty)$ but it has no inflection point.
Mar 29, 2018 at 13:30	history	edited	Robert Z	CC BY-SA 3.0	added 11 characters in body
Mar 29, 2018 at 13:27	comment	added	The Pointer		Ahh, ok. So, just to be absolutely sure, I'll repeat my question to Mundron Schmidt: 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 13:21	history	edited	Robert Z	CC BY-SA 3.0	added 11 characters in body
Mar 29, 2018 at 13:20	comment	added	Robert Z		Injectivity is not influenced by the appearance of the inflection point! If $f'\geq0$ in $(a,b)$ and $f'=0$ at a finite number of points in $(a,b)$ then $f$ is strictly increasing by the MVT.
Mar 29, 2018 at 13:16	history	edited	Robert Z	CC BY-SA 3.0	added 33 characters in body
Mar 29, 2018 at 13:15	comment	added	The Pointer		Thanks for the response, but I was more-so looking for something that directly addresses my 3 points above. Enlightening approach, nonetheless.
Mar 29, 2018 at 13:14	history	answered	Robert Z	CC BY-SA 3.0