This may seem like an elementary question, but bear with me; you'll find that it is actually quite hard. Consider the function $$f(x)=\frac{a_nx^n}{\sum_{i=0}^n a_ix^i}$$ with all $a_i\geq 0$ and $a_0=1$. Can you prove that the maximum derivative over $x\in (0,\infty)$, if it exists, is unique?

The usual elementary root counting techniques do not work since $f''(x)$ may have many sign changes. It is hard to control the number of zeros. Perhaps arguing that the numerator and denominator of $f(x)$ and $f'(x)$ are strictly monotone increasing may help.