I was about ready to move on to a second example, when one of my students asked, "Is the inflection point always going to be the midpoint of the critical points?" I acknowledged that it was a great question, but started to explain right away why I was pretty sure the answer was no, knowing that a higher odd degree function could be created to have two critical points with any variety of different behaviors in between.
What was I thinking?! Here I had a perfect opportunity, a student-initiated reason to do and pay attention to more examples, and the first thing I did was to dismiss the possibility that the conjecture was true. Halfway through my explanation I realized that I was making a mistake, but fortunately for me, my students were undeterred. Instead of dropping the point, another student refined the conjecture, and suggested, "Maybe it only happens when the function displays its maximum number of possible turning points," like a cubic function with two turning points, or a quartic function with three turning points, etc.
I really didn't know whether that conjecture was true or not, but I had also finally realized how valuable this student inquiry was. We let the question be the drive for our next few examples. We took a look at another cubic function next, fairly similar to the first, and once again the inflection point was the midpoint between the two critical points. Since our class time was running short, we decided to move on to a quartic example. We used Desmos to set up a quartic graph with sliders for the coefficients, which made it easy to manipulate and create a function with three turning points.
The quartic function we looked at was , which did turn out to be a counterexample for our class' conjecture. Class was just about over, so we didn't have time to look at more examples, but even as they were packing up, a few of my students were still throwing out possible modifiers on the conjecture.
That class period reminded me, and demonstrated to me in new ways, how valuable it can be to run with students' questions. The question about the critical and inflection points' relationships to one another had most of the class curious. It provided an "intellectual need" for more examples. It even added some suspense to the remaining examples we covered. The whole class was engaged in exploring whether the conjecture was true, or otherwise when it might be true. This in turn meant they were thinking critically about the topic, and asking questions that got beyond the plain mechanics of how to find and plot critical points and inflection points.
Seeing the value in that experience, I have tried to be more open to exploring student questions, particularly with my calculus class. They have always asked more questions than most of my other groups, but it seems they have been asking even more, as we take more time to consider them in class. I am finding that I have to be selective again, though, about which bigger questions we take time for. There are some that have more potential to lead to enhanced learning than others, and sometimes the "others" need to be left for another time for the sake of covering a new concept. But maybe my priorities still need some adjusting?
As for the question of when inflection points coincide with the midpoint of two critical points, I still don't know the answer, but we have noticed it in many examples since that class period. Maybe it's a commonly known theorem, but I don't want to Google it yet. I'm hoping to find a day yet this year for my class to explore the question to see if we can find some commonalities in the functions that behave that way. It would be a good exploratory math experience for them.