I assume you solve for when the divivative is equal to zero? In this case you would indeed have a local extrema (maxima or minima). But maybe your function behaves in such a way that it does not have maxima for finite values of the radius, and therefore the extramas you find are always minima.

For the particular case of a convex continuous real function f defined on an open convex subset C of a topological vector space, let x be a point of C. Then x is a minimum point of C if and only if 0 is a sub - gradient of f at x (0 is an element of the sub - differential of f at x). If in addition f is differentiable on C, then f '(x) = 0 if and only if x is a minimum point of f. Under some different assumptions on the convex set C (such as compactness or a more general assumption), the maximum principle for convex continuous functions on C holds. Namely the maximum of f on C is attained at at least one extreme point of C. 

I should point out that for maximum principle, C might be a convex compact subset in a locally convex space, or an unbounded closed  "generalization " of such a subset. Going back to the minimum point of a strictly convex and differentiable function f on the open (convex) subset C, then f '(x) = 0 if and only if x is a strictly global minimum point of f. In this particular case, such a minimum point is unique.