*Note: This question is based on a previous question*

I was continuing my research from last time, and I realized my question was too strict! Instead of the polynomial being strictly increasing, it only has to be only positive with the *maximum* smaller than $p(0)$. So, my new question is below:

Given $b$ and $c$ with $b,c>1$, is it possible to construct a polynomial $p(x)$, whose degree $n$ depends on neither $c$ nor $b$, such that:

$p$ is non-negative on $[0,c]$

and $b \cdot \max_{x \in [1,c] }p(x) < p(0)$? (if it can't be done, I will also accept a polynomial that satisfies the previous condition and this condition where $b=c$)