Let \(\mathscr{S}\) be the set of points inside a given equilateral triangle \(A B C\) with side 1 or on its boundary. For any \(M \in \mathscr{S}, a_M, b_M, c_M\) denote the distances from \(M\) to \(B C, C A, A B\), respectively. Define

\[

f(M)=a_M^3\left(b_M-c_M\right)+b_M^3\left(c_M-a_M\right)+c_M^3\left(a_M-b_M\right) .

\]

(a) Describe the set \(\{M \in \mathscr{J} \mid f(M) \geq 0\}\) geometrically.

(b) Find the minimum and maximum values of \(f(M)\) as well as the points in which these are attained.