Question 151
For each positive integer $n$, let $c(n)$ be the largest real number such that
$$
c(n) \leq \left| \frac{f(a) - f(b)}{a - b} \right|
$$
for all triples $(f, a, b)$ such that
* $f$ is a polynomial of degree $n$ taking integers to integers, and
* $a, b$ are integers with $f(a) \neq f(b)$.
Find $c(n)$.
$$
c(n) \leq \left| \frac{f(a) - f(b)}{a - b} \right|
$$
for all triples $(f, a, b)$ such that
* $f$ is a polynomial of degree $n$ taking integers to integers, and
* $a, b$ are integers with $f(a) \neq f(b)$.
Find $c(n)$.