In this paper, for a scalar linear differential equation with a delay, the problem of finding the coefficients of the equation and the delay value from a compact set, at which the stability margin (the maximum real part of the roots of the characteristic equation, taken with the inverse sign) is maximal, is considered. The compactness of the set guarantees the existence of a point with the maximal stability margin. The solution is based on the analysis of the roots of the characteristic equation and the analysis of the movement of the boundaries of the stability domain with their continuous transition. The transition is caused by the linear change of a parameter in the characteristic equation. The main maximum criteria that solve the problem are formulated.