Pi‐coefficients [4] provide a quantitative measure of system stability. With the goal of finding the largest stability range of the feedback gain in a system, the max‐pproblem [9] can be formulated so that its solution gives an optimal set ofpi‐coefficients. This paper simplifies the max‐pproblem by analyzing the special structure of a negative semi‐definite matrix, called theM‐matrix. To find the optimalpi‐coefficients of the general second‐order systems, seven solutions corresponding to different conditions are derived. In the examples we apply the results to a model reference adaptive system, calculate the pi‐coefficients of its error dynamics model, and use the optimalpi‐coefficient curves to determine a good sampling period.