Chapter D1.5, Problem 32E

Growth rate functions

a.    Show that the logistic growth rate function $f(P)=rP\left(1-\frac{P}{K}\right)$ has a maximum value of $\frac{rK}{4}$at the point $P=\frac{K}{2}$.

b.    Show that the Gompertz growth rate function $f(M)=-rM\ln \left(\frac{M}{K}\right)$ has a maximum value of $\frac{rK}{e}$ at the point $M=\frac{K}{e}$.