Let $f(x)=\sum\limits_{i=0}^n a_i x^i$ be a polynomial that given in implicit way, so I dont know $a_i$. I need to know if all its coefficient are positive.
A brute force way to do it is check out its derivative - if
$$
\frac{d^i f(x)}{dx^n}>0,
$$
then and $a_i>0.$
Question. Is there an elegant way to determine if the polynomial has only positive coefficients?