Chapter 13.9, Problem 16E

Shannon Diversity IndexOne 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 $H$ ?