So I've been working at this for a while and have so far been unable to find any resources on maximizing a particularly strange function that I've been trying to deal with. The function is of the form
$\alpha(x,y) = x*(1-e^{-y/x})$
$\beta(x,y) = x*f(x,y)$
$\large {f(x,y) = \frac{\sinh(\alpha(x,y))}{\cosh(\alpha(x,y)) + \frac{y}{1-y}e^{\beta(x,y)}}}$
where $x\in(0,\infty)$ and $y\in(0,1)$
It isn't too hard to show that, asymptotically, $f(x,y)$ goes to zero at either limit of x and y. Additionally, from plotting it numerically, it is clear that there is always a unique optimal value of x and y. Try as I might, I haven't found anything even close to an explicit form for the relationship between x,y and the maximal value of $f(x,y)$. I've tried Lagrange multipliers but the recursion always leads to seemingly useless statements that require me to know the value of $f(x,y)$ I am looking for. One relationship we can use is
$\large {\beta(x,y) = x\frac{\sinh(\alpha(x,y))}{\cosh(\alpha(x,y)) + \frac{y}{1-y}e^{\beta(x,y)}}}$
But that only seems to constrain the relationship between $\alpha$ and $\beta$ as functions of x and y, not telling you anything about the maximum for $f$. Any insight on how one might deal with this sort of function would be greatly appreciated, this thing has been plaguing me for a while and I'm out of ideas. I keep running into a wall where the recursive aspect makes everything hopelessly intractable. The asymptotic analysis is nice but I really want some kind of explicit formula, or at least a proof that one doesn't exist in closed form.