A common transformation when faced with a problem of this type:
$${\rm maximize} \min (f(x), g(x))$$
is to instead solve the equivalent problem
$${\rm maximize} \ \ \ z $$ subject to $$ z\le f(x); z\le g(x).$$
This can be helpful, for instance, in making the problem more tractable for some numerical optimization methods (if they are better at handling inequality constraints rather than complicated objective).
Many other transformations are possible; for instance, you can replace the inequality constraints above with equality constraints via {\em slack variables}variables. For a discussion of transformations of optimization problems, I recommend Section 4.1.3 of Boyd and Vandenberghe (free PDF here). Actually, I recommend the whole book.
