Works great. I figured it was a simple question. Thanks,
Try NumericQ:
NumericQ
ClearAll[f, min]; f[x_, a_] := x - x^2 + a/3 x^3; min[a_?NumericQ] := FindMinimum[{f[x, a], x >= 0}, x][[1]]; FindRoot[min[a] == 0, {a, 1}] (* {a -> 0.75} *)
See https://support.wolfram.com/12502 for a discussion of ?NumericQ.
?NumericQ