Chapter 5, Problem 5.28TE

Consider the beta distribution with parameters $\left(a,b\right)$. Show that

a. when $a>1$ and $b>1$, the density is unimodal (that is, it has a unique mode) with mode equal to $\frac{\left(a-1\right)}{\left(a+b-2\right)}$;

b. when $a\le 1,a\le 1$, and $a+b<2$, the density is either unimodal with mode at 0 or I or U-shaped with modes at both 0 and 1;

c. when $a=1=b$. all points in $\left[0,1\right]$ are modes.