Given a function, how to prove that it is sigmoidal in nature. So far, my approach has been to verify if the properties of sigmoidal functions hold:
1)That it is monotonic
2)That it is constrained by a pair of horizontal asymptotes
3)That it has a first derivative that is "bell" shaped
4)That it is convex for values less than 0 and concave for values more than 0
5)That a Sigmoid function and its affine compositions can posses multiple optima.
For example this is one of the functions that I am trying to verify:
Trying to prove all the properties seems like time consuming especially in exam scenarios. I suppose there is a better approach that exists and I should follow. Please suggest me what other approach could be used.