Asked by meow
Let f(x)=201+9e−3x .
What is the point of maximum growth rate for the logistic function f(x)=20/1+9e^-3x ?
Round your answer to the nearest hundredth.
(0, 2) <my choice
(5.54, 9)
(0.73, 10)
(0.73, 20)
What is the point of maximum growth rate for the logistic function f(x)=20/1+9e^-3x ?
Round your answer to the nearest hundredth.
(0, 2) <my choice
(5.54, 9)
(0.73, 10)
(0.73, 20)
Answers
Answered by
Steve
some parens help make things clearer online:
f(x)=20/(1+9e^(-3x))
f'(x) = 540e^(3x)/(9+e^(3x))^2
the greatest growth rate occurs when f' has a maximum. So,
f"(x) = 1620e^3x * (9-e^3x)/(9+e^3x)^3
since the denominator is never zero, f" is zero when
e^3x = 9
x = ln9/3 ? 0.732
you can see that the curve is steepest there:
http://www.wolframalpha.com/input/?i=plot+y%3D20%2F(1%2B9e%5E(-3x))+for+x%3D0..3
f(x)=20/(1+9e^(-3x))
f'(x) = 540e^(3x)/(9+e^(3x))^2
the greatest growth rate occurs when f' has a maximum. So,
f"(x) = 1620e^3x * (9-e^3x)/(9+e^3x)^3
since the denominator is never zero, f" is zero when
e^3x = 9
x = ln9/3 ? 0.732
you can see that the curve is steepest there:
http://www.wolframalpha.com/input/?i=plot+y%3D20%2F(1%2B9e%5E(-3x))+for+x%3D0..3
Answered by
Steve
Hmmm. f'(ln9/3) = 15
not 10 or 20
not 10 or 20
Answered by
meow
would it be (5.54, 9)?
Answered by
Steve
NO!
x = 0.73
x = 0.73
Answered by
lemion
its (0.73,10)
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