# How do you find the inflection point of a logistic function?

The answer is ##((ln A)/k, K/2)##, where ##K## is the carrying capacity and ##A=(K-P_0)/P_0##.

Don't use plagiarized sources. Get Your Custom Essay on
How do you find the inflection point of a logistic function?
Just from \$13/Page

To solve this, we solve it like any other inflection point; we find where the second derivative is zero.

##P(t)=K/(1+Ae^(-kt))##
##=K(1+Ae^(-kt))^(-1)##
##P'(t)=-K(1+Ae^(-kt))^(-2)(-Ake^(-kt))## power
##P”(t)=2K(1+Ae^(-kt))^(-3)(-Ake^(-kt))^2-K(1+Ae^(-kt))^(-2)(Ak^2e^(-kt))## product and chain rule

Now we solve:

##2K(1+Ae^(-kt))^(-3)(-Ake^(-kt))^2-K(1+Ae^(-kt))^(-2)(Ak^2e^(-kt))=0##
##2(1+Ae^(-kt))^(-1)(-Ake^(-kt))^2-(Ak^2e^(-kt))=0## cancel
##2(1+Ae^(-kt))^(-1)(Ake^(-kt))^2-k(Ake^(-kt))=0## factor out
##2(1+Ae^(-kt))^(-1)(Ake^(-kt))-k=0## cancel
##2(1+Ae^(-kt))^(-1)(Ake^(-kt))=k##
##2Ake^(-kt)=k(1+Ae^(-kt))## cancel
##2Ae^(-kt)=1+Ae^(-kt)##
##2Ae^(-kt)-Ae^(-kt)=1##
##Ae^(-kt)=1##
##e^(-kt)=1/A##
##-kt=-lnA## log rules
##t=(lnA)/k##

This gives us ##t## which we can substitute into ##P(t)##:

##P((lnA)/k)=K/(1+Ae^(-k(lnA)/k))##
##=K/(1+Ae^(-(lnA)))## log rules
##=K/(1+A/A)##
##=K/(1+1)##
##=K/2##

It’s a lot of algebra, so be very careful with factoring, cancelling, and negative signs.

## Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
\$26
The price is based on these factors:
Number of pages
Urgency
Basic features
• Free title page and bibliography
• Unlimited revisions
• Plagiarism-free guarantee
• Money-back guarantee
On-demand options
• Writer’s samples
• Part-by-part delivery
• Overnight delivery
• Copies of used sources
Paper format
• 275 words per page
• 12 pt Arial/Times New Roman
• Double line spacing
• Any citation style (APA, MLA, Chicago/Turabian, Harvard)

# Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

### Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

### Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

### Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.