Locus of the maximal turning point and the point of inflection

Suppose you have a carton that has the form of a square with sides of length a. If we want to produce a box out of it whose height is x we might deduce the following formula:

$$V_a(x)= x(a-2x)^2=a^2 x - 4 a x^2 + 4 x^3$$

I want to explore this formula with my students because it is very interesting to see that the specified points in the title of this thread are both located on two special curves. Since I don't want to say to my students here is your exercise and start working, I am reflecting on creating a good introduction that is also exciting for the students.

Since the locus has no use in an application related to this exercise I decided that I don't want to use a problemoriented introduction. The point is that every problemoriented introduction to this topic or discovery oriented process leads to application contexts which only need the y-coordinates of the specified points. So I hope that you might have an idea for an introduction to my students that is a little bit funny or exciting.

Remark: My students are able to examine functions based on their term by using the concept of the first, second and third derivative.

• You should clarify what you are looking for. For example, the point of inflection is at $\left(\frac{a}{3},V_a(\frac{a}{3})\right)$. Do you want your students to investigate this curve as $a$ varies (locus)? Somehow looking at all this in terms of $a$ seems uninteresting because $a$ just establishes a unit of length. May 1 '21 at 16:27
• The idea is to say that the students deduce the formula for $V_a$. They shall start to examine the formula by the graphs of $V_a$, which are represented in a geogebra file. I inserted in this file a scroll bar so that the students might vary a and see directly the effect on the graphs of $V_a$. The idea is then to formulate assumptions about which properties characterize these graphs. In the next lesson the students shall prove these assumptions by their own hands. May 1 '21 at 17:37