Amazing Polynomials from America

There is a smallest positive real number $a$ such that there exists a positive number $b$ such that all the roots of the polynomial $x^3 - ax^2 + bx - a$ are real. In fact, for this value of $a$ , the value of $b$ is unique. What is the value of $b$ ?

This is a problem from the 2016 AMC (12 A #24).