Let $S$ be the set of points inside a given equilateral triangle $ABC$ with side $1$ or on its boundary. For any $M \in S, a_M, b_M, c_M$ denote the distances from $M$ to $BC,CA,AB$, respectively. Define \[f(M) = a_M^3 (b_M - c_M) + b_M^3(c_M - a_M) + c_M^3(a_M - b_M).\] (a) Describe the set $\{M \in S | f(M) \geq 0\}$ geometrically. (b) Find the minimum and maximum values of $f(M)$ as well as the points in which these are attained.