Module[{f, vals},
 f[x_] :=
  0.39386402741085563` x + 0.6503344648379578` x^2 +
   0.22542329506772663` x^3 - 0.26962178731654` x^4;
 vals = c /@ Range@3;
 Sort@ArgMax[{Min[Abs[f[#1] - f[#2]] & @@@ Subsets[vals, {2}]],
    0 <= # <= 1 & /@ vals}, vals]]

(* {0.0037626, 0.613897, 0.999995} *)

How that result makes sense? f is zero at 0 and 1 at 1, and doesn't go below or above those on that interval. I'd expect smallest value on the result to be 0 and largest 1...

Oh well. I suppose ArgMax is really NArgMax when inexact coefficients are involved.

I still have to say I'm surprised by the crudeness of that result.