Question 80
Let $x_1 \lt x_2 \lt \dots \lt x_n$ be real numbers. Define the function $f : \mathbb{R} \rightarrow \mathbb{R}$
$$
f(x) = \left| \frac{(x-x_2)(x-x_3)\cdots(x-x_n)}{(x_1-x_2)(x_1-x_3)\cdots(x_1-x_n)} \right| + \cdots + \left| \frac{(x-x_1)(x-x_2)\cdots(x-x_{n-1})}{(x_n-x_1)(x_n-x_2)\cdots(x_n-x_{n-1})} \right|
$$
Prove that there exists an $i \in \{1, \dots, n-1\}$ such that for all $x \in (x_i, x_{i+1})$, we have $f(x) \lt \sqrt{n}$.
$$
f(x) = \left| \frac{(x-x_2)(x-x_3)\cdots(x-x_n)}{(x_1-x_2)(x_1-x_3)\cdots(x_1-x_n)} \right| + \cdots + \left| \frac{(x-x_1)(x-x_2)\cdots(x-x_{n-1})}{(x_n-x_1)(x_n-x_2)\cdots(x_n-x_{n-1})} \right|
$$
Prove that there exists an $i \in \{1, \dots, n-1\}$ such that for all $x \in (x_i, x_{i+1})$, we have $f(x) \lt \sqrt{n}$.