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) 0.

.

f(x) has a local extrema at , i.e.

with

between and and between and there should be monotone growth(positive or negative).

Every possible non negative combination of should be achievable.(If adding a size constraint or a constraint on the maximum difference between 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:

and .

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 , 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