Section 14.7: Max Min Problems

The Shannon index is a measure of diversity in an ecosystem. For the case of three species, it is defined as \(H = -p_1 \ln p_1 - p_2 \ln p_2 - p_3 \ln p_3\) where \(p_i\) is the proportion of species i in the ecosystem. (a) Express H as a function of two variables \(p_1\) and \(p_2\) using the fact that \(p_1 + p_2 + p_3 = 1\). (b) What is the domain of H? (c) Find the maximum value of H. For what values of \(p_1, p_2, p_3\) does it occur?