This is just my comment reposted as an answer.
The problem of which functions $f:[0,1]\to\mathbb{R}$ can be approximated by polynomials with non negative coefficients has already been studied: as theorem 2 of this paper states, a continuous function $f:[0,1]\to\mathbb{R}$ is a pointwise limit of polynomials with non negative coefficients iff $f(x)=\sum_{n=0}^\infty a_nx^n$ for some non negative sequence $a_n$ such that $\sum a_n<\infty$.