Question

PLEASE HELP ASAP!!! CORRECT ANSWERS ONLY PLEASE!!!

What is the point of maximum growth rate for the logistic function f(x) ?

Round your answer to the nearest hundredth.

Answer 1

Answer: C. (0.73, 10)

Explanation:

Maximum growth occurs at f'(x). Maximum growth rate occurs when f''(x) = 0

`f(x) = (20)/(1+9e^(-3x))`

`f'(x) = (540e^-3x)/((1+9e^(-3x))^2)`

`f''(x)=(1620e^(-6x)(-e^(3x)+9))/((1+9e^(-3x))^3)`

`0=(1620e^(-6x)(-e^(3x)+9))/((1+9e^(-3x))^3)`

`0 = (1620e^(-6x))(-e^(3x)+9)`

`1620e^(-6x)=0\\ e^(-6x)=0\\ ln\ e^(-6x)=ln\ 0\\ ln\ 0\text{\ is NOT VALID}`

`-e^(3x)+9=0\\ 9=e^(3x)\\ ln\ 9=ln\ e^(3x)\\ ln\ 9=3x\\ \\ (ln\ 9)/(3)=x\\ \\ 0.73 = x`

`f(0.73) = (20)/(1+9e^(-3(0.73)))`

= 10