CIIM 2018 Problem 4

Let $\alpha < 0 < \beta$ and consider the polynomial $f(x) = x(x-\alpha)(x-\beta)$ . Let $S$ be the set of real numbers $s$ such that $f(x) - s$ has three different real roots. For $s\in S$ , let $p(x)$ the product of the smallest and largest root of $f(x)-s$ . Determine the smallest possible value that $p(s)$ for $s\in S$ .