Let $C,D > 0$. We call a function $f : \Bbb R \to \Bbb R$ pretty if $f$ is a $\Bbb C^2$-class, $|x^3 f(x)| \leq C$ and $|xf''(x)| \leq D$.
(i). Show that if $f$ is pretty, then, given $\epsilon > 0$, there is a $x_o \geq 0$ such that for every $x$ with $|x| \geq x_o$, we have $|x^2 f'(x)| < \sqrt{2CD} + \epsilon.$
(ii). Show that if $0 < E < \sqrt{2CD}$ then there is a pretty function $f$ such that for every $x_o \geq 0$ there is $a x > x_o$ such that $|x^2 f'(x)| > E$.
This is a problem from Brazilian undergraduate math olympiad. I am not interested in solutions. What I'd like to know is where this definition came from. Does somebody know if it is related to something more advanced, or was it created just for the problem? If it is the latter, I'd like to know how this problem was created. What was the thought process? What was the inspiration? Is it possible to know?
It's just a prefabricated definition, it was written that way so that (i) and (ii) hold, it has no other use than to evaluate the skills of the participants.