Challenge Level

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.

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?

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.

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.