In this one problem we have to find values of x, y and z such that Shannon's Diversity Index (H) is maximized. $$ H = -x\ln{x}-y\ln{y}-z\ln{z} $$ being x + y + z = 1
What I've tried to do: isolate z, so that z = 1 - x - y. H then becomes f(x, y) and acquiring the partial derivatives here would allow me to find critical points. I did so, and arrived at x = y, but I can't seem to find the correct path from here, assuming that it is right. The answer to the question is 1/3.