Chapter 4.6, Problem 73E

To determine

(a)

It is to verify that,

$P\text{'}(x)=\frac{MAk{e}^{-kx}}{{(1+A{e}^{-kx})}^2}$ and $P\text{'}\text{'}(x)=\frac{MA{k}^2{e}^{-kx}(A{e}^{-kx}-1)}{{(1+A{e}^{-kx})}^3}$

To determine

(b)

To prove if $\underset{x\to -\infty }{\lim }P(x)=0$ and $\underset{x\to \infty }{\lim }P(x)=M$, therefore $P=0$ and $P=M$ are horizontal asymptotes of $P$.

To determine

(c)

It is to verify that, $P$ is increasing for all $x$.

To determine

(d)

The only inflection point of $P$ is at $\left(\frac{\ln A}{k},\frac{M}{2}\right)$. To the left of it $P$ is concave up and to the right of it $P$ is concave down.