Finding the inflection point of a sigmoid function

Hi, here's my function:

% define params
p1 = 0.0389;
p2 = 1.9158;
p3 = 162.4272;
p4 = 0.012;

% simulate the function
syms x
f = p1+(p2-p1)/(1+10^((p3-x)*p4));
fplot(f,[1 245])

Now, I would like to find the inflection point (I manually drew where I roughly expect it to be).

Here's what I was trying to do:

double(solve(f,'MaxDegree',3))

but I get this:

ans = 2.1394e+01 - 1.1370e+02i

Which means that there is no real inflection point... Where's my mistake?

Thanks in advance

Iddo

Answer 1

[Star Strider]

Try this:

p1 = 0.0389;
p2 = 1.9158;
p3 = 162.4272;
p4 = 0.012;

% simulate the function
syms x
f(x) = p1+(p2-p1)/(1+10^((p3-x)*p4));
inflpt = vpasolve(diff(f,2) == 0);                          % Calculate Inflection Point

figure(1)
fplot(f,[1 245])
hold on
plot(inflpt, f(inflpt), 'pr', 'MarkerSize',10, 'MarkerFaceColor','g')
hold off

[Iddo Weiner]

Great, thanks

[Star Strider]

As always, my pleasure.

[Karan Gill]

Iddo, did you understand why that code works?

[Iddo Weiner]

Yes, setting the 2nd derivative to equal zero will give you the inflection point

Answer 2

[Roger Stafford]

The inflection point occurs at x = p3. You can show it by using the symbolic 'diff' to find the second derivative of f with respect to x and finding the x that makes it zero. That occurs when 10^((p3-x)*p4)) is equal to 1 which forces x to equal p3.

[Iddo Weiner]

Hey Roger, thanks for your help.

I understand what you say and it's right for this case, but I would like to have a piece of code that automatically calculates the inflection point.. Looking at the official documentation it looked like this line of code is supposed to do the work:

double(solve(f,'MaxDegree',3))

But, it didn't work.. So do you see a different way to automatically generate the inflection point? Or maybe I'm not using solve() properly?

Thanks, Iddo