"Inflection " point of the sigmoid

Question

I have a very short question, this might be the wrong place to post it so I do apologise.

In the standard sigmoid function, $x \mapsto \frac{1}{1+e^{-x}}$ we have a point that intuitively should be called its inflection, namely $0$ where the second derivative is $0$. However if I think back to my analysis an inflection point must be a root also of the first derivative.

Is there a technical term for the "inflection" , second derivative root , "point where the function starts going up faster" of the sigmoid function?

Again sorry for how trivial this might be for most of you.

I have attached an image in which the blue point is the one I mean.

Answer 1

Edited:

Actually an inflection point is just a point where the second derivative changes sign irrespective of whether the first derivative is zero. The term "inflection point" does not capture the distinction between e.g. $x^3$ and $\frac{1}{1+e^{-x}}$ at $0$.