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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.

Least of All

Age 16 to 18
Challenge Level

Why do this problem ?

This problem allows students to explore the process of making mathematical conjectures based on visual intuition. The conjectures can then be proved/disproved following a straightforward exercise in finding turning points (whilst taking care over distinguishing the various cases). This is good training and practice of key mathematical skills.

Possible approach

Have a class discussion about the positions of X that might lead to a minimum of the function before anyone actually writes anything down. Consider and discuss what happens to the function as X moves along the line segment from P to Q. What conjectures do the class come up with? Is there a general agreement or are multiple conjectures stated?

After the students have found the turning points have the class reflect on whether their findings agree with the original conjecture(s) and if not why not?

Key questions

  • What is a conjecture?
  • What features of the geometric situation do you notice?
  • How do you expect the function to change as X moves from P to Q?
  • Would you expect the position of X giving a minumum value of the function to be dependent or independent of scaling, that is would this position of X be the same relative to the endpoints for a short line segments as when the line segment was stretched to make it longer

and, after doing the calculus:
  • What does the graph of the function look like as x varies?
  • Does this graph look the same whatever the length of the line segment?
  • How many turning points are there?
  • Does the nature of the turning points, and/or the position of X where the turning points occur, depend on the value of the constant a?
  • Did we miss any of the possibilities when we made our initial conjecture and if so why?
Possible extension

Consider generalisations of this type of problem.

Possible support

Students might struggle with the concept of a conjecture as a precise mathematical statement which can be proved or disproved following further investigation. You might like to suggest these possibilities and to vote on them
  • I think that the location of the minimum does not depend on the value of a
  • I think that the location of the minimum does depend of the value of a
  • I think that the minimum values are always located in the centre of the line.