Chapter 13.9, Problem 16E

Shannon Diversity Index One way to measure species diversity is to use the Shannon diversity index H. If a habitat consists of three species. A, B, and C, then its Shannon diversity index is

$H=-x\ln x-y\ln y-z\ln z$

where x is the percent of species A in the habitat, y is the percent of species B in the habitat, and z is the percent of species C in the habitat. Use the fact that

$x+y+z=1$

to show that the maximum value of H occurs when $x=y=z=\frac{1}{3}$. What is the maximum value of H?