Question 115
Let $a$ and $b$ be real numbers, $a \lt b$, and let $f$ be a polynomial of degree 3, with real coefficients, so that the polynomials $f - a$ and $f - b$ have only real roots.
Prove that the set of the real numbers $x$ such that $a \lt f(x) \lt b$ is the disjoint union of three open intervals, and the length of one of these intervals is the sum of the lengths of the other two.
Prove that the set of the real numbers $x$ such that $a \lt f(x) \lt b$ is the disjoint union of three open intervals, and the length of one of these intervals is the sum of the lengths of the other two.