Yes, that's only possible if both derivatives are nonnegative, for obvious reasons. Much trickier to do, too. It seems that we have to consider several distinct cases depending on values of u'(0) and u'(1). For example, when they are both greater than 1, we can construct the necessary function out of x^a and 1(1x)^b. Can't think of a good function that works for all values.
