MathNet 2021 — Part 03 Question 5

Let $a, b$ be real numbers and function $f(x) = x^3 + a x^2 + b x$. If there exist three real numbers $x_1, x_2, x_3$ satisfying $x_1 + 1 \le x_2 \le x_3 - 1$ and $f(x_1) = f(x_2) = f(x_3)$. Find the minimum of $|a| + 2|b|$.