This paper studies a mixedH2/H∞control problem with internal stability by means of memoryless state feedback, for smooth nonlinear control systems. The problem consists of combining the performance requirements of quadratic optimal controllers with the robustness properties ofH∞controllers. It is shown that under appropriate assumptions, the problem is solvable if a pair of cross-coupled Hamilton-Jacobi-Isaacs equations allows smooth solutions. Our approach to the mixedH2/H∞nonlinear control problem is drawn from the concepts of non-zero sum differential game, LaSalle's Invariance Principle and completion of squares. Analogous results are also developed for discrete-time nonlinear systems, by using the Taylor expansion argument. Finally, all the results obtained in affine systems are generalized to non-affine nonlinear systems.