A considerable number of problems in operations research and statistics have the following form maximize ∫ f[x, y(x)] dx subject to ∫ g[x, y(x)] dx = constant with respect to bounded y(x). We give a necessary and sufficient condition for a maximizing function under fairly weak restrictions and prove its existence. The solution is applied to a general version of B. O. Koopman's search problem, and to the Neyman, Pearson lemma of statistics. We also show that in the discrete version of this problem, where x is replaced by an index and sums replace integrals, our condition is sufficient but not necessary and give, as illustration of the sufficiency, a solution to an assignment problem.