Find the logistic function f with the given properties HINT

Find the logistic function f with the given properties. HINT [See Example 1.]

f has limiting value 7 and passes through (0, 1) and (1, 2).

Solution

general form of logistic equation is f(x) =N_/(1+ae^-bx), where N is limiting value snd a,b are constants

f has limiting value 7

f(x) =7_/(1+ae^-bx)

passes through (0, 1)

1=7_/(1+ae⁰)

1=7_/(1+a)

1+a=7

a=6

f(x) =7_/(1+6e^-bx)

passes through (1, 2).

2 =7_/(1+6e^-b)

(1+6e^-b)=7_/2

6e^-b=5_/2

e^-b=5_/12

e^b=12/5

b=ln(12/5)

f(x) =7_{/(1+6e^-(ln(12/5))x)} is the  logistic function