Question 85
a) Show that, if $I \subset \mathbb{R}$ is a closed bounded interval, and $f: I \to \mathbb{R}$ is a non-constant monic polynomial function such that $\max_{x \in I} |f(x)| \lt 2$, then there exists a non-constant monic polynomial function $g: I \to \mathbb{R}$ such that $\max_{x \in I} |g(x)| \lt 1$.
b) Show that there exists a closed bounded interval $I \subset \mathbb{R}$ such that $\max_{x \in I} |f(x)| \ge 2$ for every non-constant monic polynomial function $f: I \to \mathbb{R}$.
b) Show that there exists a closed bounded interval $I \subset \mathbb{R}$ such that $\max_{x \in I} |f(x)| \ge 2$ for every non-constant monic polynomial function $f: I \to \mathbb{R}$.