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## 'Curve Fitter' printed from http://nrich.maths.org/

I wish to find a function which passes through the origin and only
has turning points at $(1, 2)$ and $(2,1)$, as in the following
sketch:

What sorts of equations would be good candidates for having a graph
of this form? Can you definitely rule out certain types of equation
from having a graph like this?

Although this curve might look like a cubic equation, it actually
cannot be a cubic equation. Prove that this is the case.

Next, can you explicity find an equation which exactly satisfies
these conditions and looks similar to the graph shown (i.e. with no
other turning points anywhere and the same asymptotic behaviour)?
How might you assess the effectiveness of the 'similarity'?