mathematical optimization – Continuous Min-Max problem

Being new to Mathematica, I tried my best to find some biult-in functions or guides on how to solve the classical min-max problem

$$min_{x} max_{k} f(x,k,params)$$

with some additional variables $$params$$ and some simple constraints on the variables (e.g., $$xin (x_{min},x_{max})$$ and $$kin (k_{min},k_{max})$$) in the Mathematica language. Finding none (giving a link would be much appreciated), my approach was to first define function computing
$$max_{k} f(x,k)$$
e.g.,

``````fMax(x_,params_) :=
FindMaximum({f(x,k,params_), k > kmin, k < kmax}, {k, kinit});
``````

with a parameter $$x$$ and then minimize `fmax`, e.g.,

``````fMinMax(x_,params_) :=
FindMinimum({fMax(x_,params_), x > xmin, x < xmax}, {x, xinit});
``````

However, the following error is consistently raised.

``````FindMaximum::nrnum: The function value -((9.27923*10^11-2.95367*10^10 p)/(5.15531*10^17+1.64099*10^16 p)) is not a real number at {k} = {10.}.
``````

although upon evaluating the function at that given point, the value is indeed real. I would be glad for any help. To give the full setting $$f$$ amounts to

$$f(x,k,a,b,alpha) = frac{frac{kpi}{b} cosh left(frac{kpi}{b} (a-alpha)right) + x sinh left(frac{kpi}{b} (a-alpha)right)}{frac{kpi}{b} cosh left(frac{kpi}{b} (a+alpha)right) + x sinh left(frac{kpi}{b} (a+alpha)right)}$$

where $$a,b,alpha$$ are positive parrameters such that $$a>alpha>0,b>0$$.