The user of RSM techniques gains considerable insight into the nature of the stationary point and the nature of the underlying response surface from estimates of the eigenvalues of the matrix of pure and mixed quadratic regression coefftcients,B. This article presents and illustrates methodology for constructing conservative confidence limits on the eigenvalues of this matrix as well as the mean response at a constrained optimum. When a confidence interval about an eigenvalue ofBincludes zero, a change in the strategy of analysis is required; one is suggested and illustrated.