TITLE: Looking for logistic function that passes through the origin and is asymptotic to y = k
QUESTION [0 upvotes]: I need to find a function with the following properties:
For all non-negative x,

F(0) = 0
lim x=> inf, F(x)=> k
F'(0) = 1
F"(x) < 0

I believe that the solution will take the form of a logistics function, but its been decades since I finished my math studies.

REPLY [0 votes]: I think $F(x)=\arctan x$ will do fine.
$F(0) = \arctan 0 = 0$
$\displaystyle \lim_{x\to +\infty} F(x)=\frac{\pi}{2}$
$F'(0) = \left.\frac{1}{1+x^2}\right|_{x=0} = 1$
$F''(x)=\left(\frac{1}{1+x^2}\right)'= -\frac{2x}{(1+x^2)^2}< 0, \quad x>0 $
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