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Why do this problem?

This problem allows for straightforward algebraic proof, a simpler, but more sophisticated, proof using geometrical reasoning or a combination of the two. The problem involves basic ideas in calculus and will naturally fit into a scheme of work for calculus.

Possible approach

The problem can be used as students are beginning to learn the terminology of turning points and differentiation of polynomials. If they have not yet encountered points of inflection you can simply define these as places where the tangent crosses the curve, which corresponds algebraically to the second derivative changing sign.
 
Ask that students neatly draw axes with rules before starting their sketches. After a few minutes exploration you might suggest that students try to sketch examples of curves of the form M, m, I, MIm, IMI, mIMm, ImIMI, with I, M and m denoting points of inflection, local maxima and local minima respectively.
 
More advanced students will wish to include asymptotes and other interesting curves. This should be encouraged, but restrict the investigation to smooth curves.
 
Conjectures to aim for might be of the form 'There is always a point of inflection between a maximum and a minimum' or 'There is always a turning point between two points of inflection'. Conjectures should be clearly stated. Better students might wish to improve their conjectures to forms such as 'There is always a point of inflection or an asymptote between a maximum and a minimum' of 'For polynomials there is always a point of inflection between a maximum and a minimum'.
 
There are two levels of sophistication in the proofs. Weaker students might simply test out their conjectures on many examples. Stronger students will go for an algebraic solution whilst the most sophisticated thinkers will use geometrical reasoning to avoid any algebra.
 
To conclude the lesson, students can refine their conjectures if necessary and share them, along with the justification or proof they have constructed.
 

Key questions

How can you spot a turning point/point of inflection geometrically (i.e. on the graph)?
 
How can you find a turning point/point of inflection algebrically?
 
For a cubic polynomial, what sort of polynomial is the first derivative? Second derivative?
 
Will a cubic polynomial always have: a maximum?  a minimum? a point of inflection?
 

Possible extension

Extension is included in the question.
 
You could also ask that students see for which of the examples M, m, I, MIm, IMI, mIMm and ImIMI they can provide an algebraic example.

Possible support

You could simply suggest that students try to show that between a maximum and a minimum there will always be a point of inflection. They could try this out on several cubic polynomials, giving practice in differentiation and use of the formula for the solution of quadratic equations.