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)
its (0.73,10)
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
Hmmm. f'(ln9/3) = 15
not 10 or 20
would it be (5.54, 9)?
NO!
x = 0.73
To find the point of maximum growth rate for the logistic function f(x) = 20/ (1 + 9e^(-3x)), we need to find the derivative of the function and locate where it equals zero.
Step 1: Find the derivative of f(x):
We have the function f(x) = 20/ (1 + 9e^(-3x)). To find its derivative, we use the quotient rule:
f'(x) = (0 * (1 + 9e^(-3x)) - 20 * (-3e^(-3x))) / (1 + 9e^(-3x))^2
Simplifying this expression, we have:
f'(x) = 60e^(-3x) / (1 + 9e^(-3x))^2
Step 2: Set the derivative equal to zero and solve for x:
To find the point where the growth rate is maximum, we need to find the x-value where the derivative is zero. So we set f'(x) = 0:
60e^(-3x) / (1 + 9e^(-3x))^2 = 0
Since the numerator can't be zero, we can ignore it. We focus on the denominator:
(1 + 9e^(-3x))^2 = 0
Taking the square root of both sides:
1 + 9e^(-3x) = 0
Now, isolate e^(-3x) term:
9e^(-3x) = -1
Divide both sides by 9:
e^(-3x) = -1/9
Step 3: Solve for x:
To eliminate the exponent, we take the natural logarithm (ln) on both sides:
ln(e^(-3x)) = ln(-1/9)
Simplifying:
-3x * ln(e) = ln(-1/9)
Since ln(e) equals 1, we get:
-3x = ln(-1/9)
Solving for x:
x = (1/3) * ln(-1/9)
Now, it's important to note that the natural logarithm of a negative number is undefined. Therefore, there's no real solution for x in this case.
As a result, none of the given choices (0, 2), (5.54, 9), (0.73, 10), or (0.73, 20) are correct.