Given a polynomial $y : \mathbb{R} \mapsto \mathbb{R}$ of degree $p$: $$ y(x) = \sum_{k=0}^p c_k\, x^k,$$ can a random set of coefficients $\{c_0, \cdots ,c_p\}$ be generated such that $y$ is monotonically increasing for all $x \in [0,1]$?
Alternatively, can a probabilistic bound on the monotonicity of $y$ be shown? For example, if $n$ tests of the nonnegativity of the slope of $y$ are made at different locations $\{x_1,\cdots, x_n\}$, can an upper bound be placed on the probability that $y$ is not monotonically increasing?
Finally, if $\epsilon$ is this bound, then can $\epsilon$ be written as a function of $n$ and $p$?