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

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 $a>alpha>0,b>0$.