Chapter 11.1, Problem 41E

Suppose that $0<y_0<N$. Let $b=\frac{\left(N-y_0\right)}{y_0}$, and let $y\left(x\right)=\frac{N}{\left(1+b{e}^{-kx}\right)}$ for all x. Show the following.

a. $0<y\left(x\right)<N$ for all x.

b. The line $y=0$ and $y=N$ are horizontal asymptotes of the graph.

c. $y\left(x\right)$ is an increasing function.

d. $\left(\frac{\ln b}{k},\frac{N}{2}\right)$ is a point of inflection of the graph.

e. $\frac{dy}{dx}$ is a maximum at $x_0=\frac{\ln b}{k}$.