Question 43
For a polynomial $P \in \mathbb{R}[x]$, let $f(P) = n$ if $n$ is the smallest positive integer such that
$$
(\forall x \in \mathbb{R}) \underbrace{(P(P(\dots P(x))\dots))}_{n} \gt 0,
$$
and $f(P) = 0$ if such an integer $n$ does not exist. Does there exist a polynomial $P \in \mathbb{R}[x]$ of degree $2014^{2015}$ such that $f(P) = 2015$?
$$
(\forall x \in \mathbb{R}) \underbrace{(P(P(\dots P(x))\dots))}_{n} \gt 0,
$$
and $f(P) = 0$ if such an integer $n$ does not exist. Does there exist a polynomial $P \in \mathbb{R}[x]$ of degree $2014^{2015}$ such that $f(P) = 2015$?