Who knows how to find the maxima for this function?
- $$f(x)=2/(1+(x^{d/u})^{(x-u)(x+1)^{a}}$$
- $a = -\ln(-((u - ku)\ln((ku)^{d/u}))/\ln(2/z - 1))/(ln(ku + 1))$
where $d, k, u, z$ are constants and
- $u>1, 1<z<2, 1/u<k<, d>0$
You can experiment with the constants here: https://www.desmos.com/calculator/5pgpbqywvj
An approximation I have found through randomizing the constants and doing curve fitting after using newtown's method looks like this:
$$(max_x) = 1.40507050224083 + 0.625438006773909 z log(u) + 0.174672097259909 z log(k)^{2} log(u) + 0.124047911240367 uz/(0.439620582483426 + d - d*k^{2}) - 0.690708328543431z$$
However, it is not accurate enough for my needs. In the image below the red dot represents max x. Overall it's the most accurate one but misses the mark for some combinations of constants. The curve looks quite predictable, and I think there should always be maxima within 1 < max x < u.
If there is an easier curve I can use which goes through (0,0), (1,1), (ku, z) and (u, 1) while looking like a skewed distribution or trapeze, then that might also be acceptable. Ideally, it would be great if the curve always had the same maximum, eg f(max x) = 2