27. Consider the beta distribution with parameters

(a, b). Show that

(a) when a ≥ 1 and b ≥ 1, the density is unimodal (that is, it has a unique mode) with mode equal to (a - 1)/(a + b - 2);

(b) when a ≥ 1, b ≥ 1, and a + b < 2, the density is either unimodal with mode at 0 or 1 or U-shaped with modes at both 0 and 1;

(c) when a ≤ 1 ≤

b, all points in [0, 1] are modes.