## Create a special function

Hello everyone. I have a rather unusual question and thank you very much in advance.
For an economics paper I am trying to create a function f(x) with four additional parameters:
The function should fulfill following aspects:

f(x) $\geq$ 0.

$\lim_{x\to\infty}f(x) = \gamma$.

$f(0) = \delta$

f(x) has a local extrema at $\psi$, i.e. $f'(\psi) = 0$

with $f(\psi) = \mu$

between $\delta$ and $\psi$ and between $\psi$ and $\gamma$ there should be monotone growth(positive or negative).

Every possible non negative combination of $\gamma,\ \delta,\ \psi,\ \mu$ should be achievable.(If adding a size constraint or a constraint on the maximum difference between $\gamma,\ \delta,\ \mu$ helps, thats ok as well.
There need not be a analytical solution, numerical would be quit satisfactory.
So far I have been experimenting with something like:
$(ax +b)*e^(-cx) + d$ and $\frac{ax +b}{x^2*c + 1}
+d$
.

Due to my economics back ground I also tried to take inspiration from the Nelson Siegel formula:

but without much success.
My problem is, that I can't seperate the position of the extreme from the value of $\psi$, so I can not move a extrema horizontally.
If anyone has any idea, as two what constructions could be used, I would be very grateful.
Thanks a lot and have a nice day