I saw that the area where the tangent line can be drawn is divided into the corresponding divided area as shown in the picture below based on the tangent line drawn from the inflection point of the cubic function.
In this picture,
green area: We can draw one tangent line.
pink area: We can draw three tangent lines.
And after studying more, I found that this theorem is also valid for quadratic functions and quartic functions. So I wondered if this theorem is valid for any function of any degree. However, I don't know how to prove it from the beginning. How can I prove this?
Etit: I thought of a function with degree n, which is an extreme case, and has many convex parts like this. I wondered if it was possible to generalize the number of regions where tangent lines can be drawn, even in such a complex case. If possible, I posted the question here in hopes of getting some hints.