We prove that the Strong Axiom of Revealed Preference tests the existence of a strictly quasiconcave (or strictly concave, strictly monotone) utility function generating finitely many demand observations. This sharpens earlier results of Afriat, Diewert, and Varian that tested (“nonparametrically”) existence of a piecewise linear utility function. For finite data sets, one implication of our result is that even some weak types of rational behavior—maximization of pseudotransitive or semitransitive preferences—are observationally equivalent to maximization of continuous, strictly concave, and strictly monotone utility functions. And for infinitely many observations, our result is the basis of several new rationality theorems.