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Folium of Descartes

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

Witch of Agnesi

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

Discrete Trends

Find the maximum value of n to the power 1/n and prove that it is a maximum.

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.