Question

Let f(x)=241+3e−1.3x . What is the point of maximum growth rate for the logistic function f(x) ? Round your answer to the nearest hundredth.

Answer 1

x = 0.82

Explanation:

I am assuming your logistic function is

`f(x) = (24 )/(1 +3e^(-1.3x) )`

The graph of the function is asymmetric, the maximum value is 24, and the point of maximum growth is at

y = 24/2

So, we can set y = 24/2 and solve for x.

`(24 )/( 2) = (24 )/(1 +3e^(-1.3x) )`

The numerators are equal, so the denominators must be equal.

`2 = 1 +3e^(-1.3x)`

`1 = 3e^(-1.3x)`

`(1 )/(3)=e^(-1.3x)`

log3 = -1.3x

-0.4771 = -1.3x

x = 0.4771/1.3

x = 0.82

The point of maximum growth is at x = 0.82.

You can see the logistic curve and the point of maximum growth in the image below.