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) $\displaystyle \geq$ 0.

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

$\displaystyle f(0) = \delta$

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

with $\displaystyle f(\psi) = \mu$

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

Every possible non negative combination of $\displaystyle \gamma,\ \delta,\ \psi,\ \mu$ should be achievable.(If adding a size constraint or a constraint on the maximum difference between $\displaystyle \gamma,\ \delta,\ \mu$ 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:
$\displaystyle (ax +b)*e^(-cx) + d$ and $\displaystyle \frac{ax +b}{x^2*c + 1}

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 $\displaystyle \psi$, 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