Existence, Uniqueness and Counterexamples · Medium problem

For $f(x) = x^3 - 4x + 1$, the values at $x = -3, -2, -1, 0, 1, 2, 3$ are $-14, 1, 4, 1, -2, 1, 16$. By sign changes alone, at least how many real roots must $f$ have?