Suppose one observes a process Y on the unit interval, where dY = ƒ + n-1/2dW with an unknown function parameter ƒ, given scale parameter n ≥ 1 ("sample size") and standard Brownian motion W. We propose two classes of tests of qualitative nonparametric hypotheses about ƒ such as monotonicity or convexity. These tests are asymptotically optimal and adaptive with respect to two different criteria. As a by-product we obtain an extension of L'evy extasciiacutes modulus of continuity of Brownian motion. It is of independent interest because of its potential applications to simultaneous confidence intervals in nonparametric curve estimation
