## Problem

While working on part (c) of Can you find… curvy cubics edition, which asked: “Can you find a cubic curve that has a local minimum when $x=-1$?”, two students had the following conversation:

A: I can find a cubic with a stationary point at $x=-1$ by having two of the $x$-axis intersection points at $x=-2$ and $x=0$, because the stationary point is half-way between the two intersection points. I need a third intersection point, too, not between these, so I may as well choose $x=4$, so my cubic is $y=-x(x+2)(x-4)$ (with a minus sign so the cubic is the right way up).

B: I’m not sure that’s right; is the stationary point really half-way between the $x$-axis intersection points?

Can you resolve their debate?

If student A is right in this case, does this approach always work for finding a cubic with a given stationary (turning) point, or only sometimes?

If student B is right in this case, does student A’s approach ever work?