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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.
Patterns of Inflection
Age 16 to 18
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.
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