  We consider the problem of inference on a regression function at a point when
the entire function satisfies a sign or shape restriction under the null. We
propose a test that achieves the optimal minimax rate adaptively over a range
of H\"{o}lder classes, up to a $\log\log n$ term, which we show to be necessary
for adaptation. We apply the results to adaptive one-sided tests for the
regression discontinuity parameter under a monotonicity restriction, the value
of a monotone regression function at the boundary and the proportion of true
null hypotheses in a multiple testing problem.
