Mathematica has a built-in function ArgMax for functions over infinite domains, based on the standard mathematical definition.

The analog for finite domains is a handy utility function. Given a function and a list (call it the domain of the function), return the element(s) of the list that maximize the function. Here's an example of finite argmax in action: http://stackoverflow.com/questions/471029/canonicalize-nfl-team-names/472213#472213

And here's my implementation of it (along with argmin for good measure):

(* argmax[f, domain] returns the element of domain for which f of 
   that element is maximal -- breaks ties in favor of first occurrence. *)
SetAttributes[{argmax, argmin}, HoldFirst];
argmax[f_, dom_List] := Fold[If[f[#1]>=f[#2], #1, #2]&, First[dom], Rest[dom]]
argmin[f_, dom_List] := argmax[-f[#]&, dom]

First, is that the most efficient way to implement argmax? What if you want the list of all maximal elements instead of just the first one?

Second, how about the related function posmax that, instead of returning the maximal element(s), returns the position(s) of the maximal elements?