For simple problems Maximize and Minimize can symbolically handle problems with parameters. Here's an example from the documentation

In[1]:= Maximize[a x^2 + b x + c, x]

Out[1]= {\[Piecewise] {
   {c, (b == 0 && a == 0) || (b == 0 && a < 0)},
   {-((b^2 - 4 a c)/(4 a)), (b > 0 && a < 0) || (b < 0 && a < 0)},
   {\[Infinity], \!\(
      TagBox["True",
       "PiecewiseDefault",
       AutoDelete->False,
       DeletionWarning->True]\)}
  }, {x -> \[Piecewise] {
     {0, (b == 0 && a == 0) || (b == 0 && a < 0)},
     {-(b/(2 a)), (b > 0 && a < 0) || (b < 0 && a < 0)},
     {Indeterminate, \!\(
        TagBox["True",
         "PiecewiseDefault",
         AutoDelete->False,
         DeletionWarning->True]\)}
    }}}

NMinimize[{(5 + x^3.5)/x, x > 0}, x]
(*  Out:   {5.742347492053465`,{x\[Rule]1.2190136481282579`}}  *)

For negative arguments your function becomes complex valued, so this should be excluded from the search. And the respective constrains refer to the argument and not to any parameters.