Calculus meets Functions

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?

What do we need to do to check who is right in this case?

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?

What exactly is student A’s approach? Can you describe it precisely?

Answering this first will ensure that we agree about what we are trying to show!

How could we show whether this approach always works?

• What methods are there for proving whether something is always, sometimes or never true?

If student B is right in this case, then how might we go about proving whether student A’s approach ever works?

• Can you think of a visual argument? You might consider the graphs of a cubic function and its derivative, and ask how they relate to each other.
• Can you think of an algebraic argument?
• How do these (or other) arguments compare?