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The critical points of a function $f(x)$ are candidates for local extrema, i.e., if a function changes from increasing to decreasing, or vice versa, it must happen at a critical point.

Is there an analogous term for candidates for inflection points? The mathematical definition is analogous - if a function changes from positive concavity to negative concavity at a point, or vice versa, it must happen at a point $a$ in the domain of $f$ where $f''(a)=0$ or where $f''(a)$ does not exist - but what do we call such a point (if we call it anything at all)?

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    $\begingroup$ I'm not sure whether this is a question specific to teaching or something that should be asked over on Mathematics StackExchange. I suppose it could work on either site. $\endgroup$ – J W Feb 12 at 9:33
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    $\begingroup$ Personally, I'm unaware of a specific term. I just use "candidate for inflection point" and notice that students often assume that $f''(a)=0$ guarantees an inflection point, despite the example of $f(x)=x^4$ at $x=0$. $\endgroup$ – J W Feb 12 at 9:37
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    $\begingroup$ I also call these "potential" or "candidate" points of inflection. But also point out what you have, that these candidate inflection points are critical points of $f'$ $\endgroup$ – Carser Feb 12 at 12:30
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    $\begingroup$ They aren't quite "critical"... maybe they are only "important"? :P $\endgroup$ – Xander Henderson Feb 13 at 1:54
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    $\begingroup$ @XanderHenderson, a candidate to inflection point ($f''(x) = 0$) isn't necessarily a critical point ($f'(x) = 0$). $\endgroup$ – vonbrand Feb 26 at 16:30
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The calculus textbook I'm currently using (Burzynski, Applied Calculus for Business, Life and Social Sciences, XYZ Textbooks) uses the term "hypercritical point". A quick web search indicates that a few other sources use it as well.

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    $\begingroup$ Perhaps you might want to give more information about the textbook (title, full name of author, publisher, year published). $\endgroup$ – Joel Reyes Noche Apr 8 at 5:18
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    $\begingroup$ Thanks! Updated my answer. $\endgroup$ – TomKern Apr 8 at 19:20
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The term Hergert Numbers is sometimes used in my specific region of the US for the values of $x$ where $f''(x) = 0$ or $f''(x)$ is undefined. This is in reference to Rodger Hergert, an Illinois community college professor who sometime in the 1990s became very frustrated with the fact that these numbers had no good name.

But it doesn't really matter what you call them as long as you are precise about it and distinguish them from inflection points, so I recommend the whimsical option where you name them after a current math educator -- call the points you are talking about Hergert points and the x-values Hergert numbers.

If a student asks who Hergert is, you can grin and truthfully say "some guy like me who was annoyed that these didn't have a name."

If you do adopt the terminology, he'll send you a shirt, too. As of the time of this writing, the directions to get the shirt are at hergertnumbers.org under "If you would like your own Hergert Numbers t-shirt."

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    $\begingroup$ Hilarious. I typically call them "potential points of inflection". $\endgroup$ – James S. Cook Apr 8 at 15:59

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