Asymptotically close to the Nth-order multicritical point of an N-phase system, there are N-1 sum rules involving the mean-field susceptibilities measured in each of the coexisting phases. These sum rules provide the experimentalist a convenient and stringent test of the theory. In particular, they facilitate the detection of nonclassical effects, especially valuable for N>3 where the fluctuation effect is dominated by the classical contribution for three-dimensional systems.