Consider the following implicit function defined using FindRoot and I am looking for its maximum.

 f[a_] := Sqrt[a] x /. FindRoot[x^4 + a x - 1 == 0, {x, 5}]
 Plot[f[a], {a, 0, 10}]
 FindMaximum[f[a], {a, 0}]

Mathematica does find the solution after some error messages:

FindRoot::nlnum: The function value {624. +5. a} is not a list of numbers with dimensions {1} at {x} = {5.}.

ReplaceAll::reps: {FindRoot[x^4+a x-1==0,{x,5}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing

I want to get rid of these error messages, because if I make the implicit function more complicated, Mathematica will attempt to evaluate the function without the parameter value, resulting in very long calculations. Your help is very much appreciate.