Chapter 7.5, Problem 54E

Shannon Diversity Index One way to measure species diversity is to use the Shannon diversity index H. A habitat consists of three species A. B. and C, and 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 $x+y+z=1$

(the sum of the three must equal $100\%$

) to show that the maximum value of H occurs when

$x=y=z=\frac{1}{3}$

What is the maximum value of H?