A couple of weeks after that lesson took place, I found a day for the class to just explore that question - when is the inflection point the midpoint of the critical points? We split the board into two sections and began recording functions that satisfied that condition and ones that didn't. My students set to work with their graphing calculators to create and explore functions. Here are some questions that arose during their work:

- What do you have to do to get two turning points?
- Is it just true for all cubic functions?
- Is it possible to find a quartic, or quintic function with an inflection point at the midpoint of two critical points?

Eventually, most of the class gave up on quartic and higher degree functions, and the focus shifted mainly to cubics. Several times a cubic function was added to the "doesn't fit" column, but after double checking these we found that all of them had inflection points directly between their midpoints. So, by the end of the class period, we had a white board full of cubic functions that showed this critical-inflection point relationship, and no counterexamples. We were inclined to assume it was true for all cubic functions.

In retrospect, it seems like a pretty obvious result, at least as far as the

*x*-values are concerned. A cubic function's first derivative is quadratic, and the vertex of a parabola (where the zero of the second derivative will occur) is always halfway between its zeros. I wasn't certain about the*y*-values, though. To be sure, I wrote up a proof (here's a link to that proof). I began with a generic, standard form cubic function and carried out the differentiation to find the first and second derivatives, and then used algebra to find the zeros of those derivatives and the accompanying*y*-values for the function.
Unfortunately, I didn't have a class period to spend working through a proof of this result with my class. The work is pretty accessible for students who have had Algebra II or more and some basic Calculus -- derivatives using the power rule, quadratic formula, and some binomial theorem for plugging numbers back in to the original cubic function. I made my proof available to my class for anyone who was interested. They were satisfied to know that their hypothesis was true at least for cubics, but they were still making conjectures about when the same relationship might be true in quartics and higher degree functions, which is great.

Maybe I didn't word it well, but when I did a Google search to see if there was anything out there about this relationship between critical and inflection points, I didn't find much. I found at least one exploration worksheet that guided students to notice the relationship for some specific examples, but not much else. Does anyone out there know if this is a commonly known relationship? Maybe it's too "obvious" to be worthy of mention in most textbooks. I didn't notice it right away.

I'm still curious, and so are my students, if this ever happens in higher degree functions. And if so, when? If anyone can point me in the direction of a resource that might help me learn more about it, I'd be grateful. Thanks!