Question 142
A polynomial is called "monic" if the coefficient of its highest-degree monomial equals $1$. Let $a \lt b$ be real numbers, and let $P$ be a non-constant monic polynomial such that $\max_{x \in [a, b]} |P(x)| \lt 2$.
Prove that there exists a polynomial $Q$, monic and non-constant, such that $\max_{x \in [a, b]} |Q(x)| \lt \frac{1}{2017}$.
Prove that there exists a polynomial $Q$, monic and non-constant, such that $\max_{x \in [a, b]} |Q(x)| \lt \frac{1}{2017}$.