Chapter 7.4, Problem 27E

a.

To determine

Prove that the 1^st and 2^nd derivatives of given logistic function are $P^{\prime }\left(t\right)=\frac{MAk{e}^{-kt}}{{\left(1+A{e}^{-kt}\right)}^2}$ and $P^{″}\left(t\right)=\frac{MA{k}^2{e}^{-kt}\left(A{e}^{-kt}-1\right)}{{\left(1+A{e}^{-kt}\right)}^3}$ .

b.

To determine

To Proof: The horizontal asymptotes are $\underset{\lim t\to \infty }{P\left(t\right)}=0\text{ and }\underset{\lim t\to \infty }{P\left(t\right)}=M$ .

c.

To determine

Proof: P is increasing for all value of t.

d.

To determine

To Prove: that the inflection point of P is at $P\left(\frac{\ln A}{k},\frac{M}{2}\right)$ .