Challenge Level

This problem is a good place to put into action the analogies for calculus explored in Calculus Analogies, so you might wish to consider that problem first.

Draw sets of coordinate axes $x-y$ and sketch a few simple smooth curves with varying numbers of turning points.

Mark on each curve the approximate locations of the maxima $M$, the minima $m$ and the points of inflection $I$.

In each case list the order in which the maxima, minima and points of inflection occur along the curve.

What patterns do you notice in these orders? Make a conjecture.

Test out your conjecture on the cubic equations

$$

3x^3-6x^2+9x+11\quad\mbox{ and } 2x^3-5x^2-4x

$$

Prove your conjecture for any cubic equation.

Extension: Consider the same problem for polynomials of order 4 or greater.