The Shannon index (sometimes called the Shannon-Wiener index or Shannon-Weaver index) is a measure of diversity in an ecosystem. For the case of three species, it is defined as

H = -p₁ ln p₁ - p₂ ln p₂ - p₃ ln p₃
where pi is the proportion of species i in the ecosystem.
(a) Express H as a function of two variables using the fact
that p₁ + p₂ + p₃ = 1.
(b) What is the domain of H?
(c) Find the maximum value of H. For what values of p₁, p₂, p₃ does it occur?