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Patterns of Inflection

Stage: 5 Challenge Level: Challenge Level:1

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.