Question

Let f (x) = 15/ 1+4e^-0.2x

What is the point of maximum growth rate for the
logistic function f(x)? Show all work.
Round your answer to the nearest hundredth.
Hint: Your answer should be an ordered pair.​

Answer 1

(6.931, 7.5)

Explanation:

The point of maximum growth of a logistic function is always halfway between the horizontal asymptotes. Here, those are y=0 and y=15, so the point of interest is where f(x) = 15/2:

15/2 = 15/(1+4e^(-.2x)) . . . . . use 15/2 for f(x)

2 = 1 +4e^(-.2x) . . . . . . . . . . . match denominators

1/4 = e^(-.2x) . . . . . . . . . . . . . subtract 1, divide by 4

ln(1/4)/-0.2 = 5·ln(4) = x ≈ 6.93147 . . . . . . take natural logs, evaluate

The ordered pair (x, f(x)) is (6.93147, 7.5).