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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.

### Quick Route

What is the quickest route across a ploughed field when your speed around the edge is greater?

# Patterns of Inflection

##### Stage: 5 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 you 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.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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