Abstract
The general extremum theory essentially uses properties of operator derivatives. As an example we consider a system described by a nonlinear elliptic equation. In this system with large values of the nonlinearity parameter and the domain dimension the control-state mapping is not Gâteaux differentiable. For this reason one cannot immediately differentiate the optimality criterion and establish the necessary optimality conditions by classical methods. However the mentioned mapping is extendedly differentiable. This allows one to obtain optimality conditions imposing no constraints on system parameters. Concluding the paper, we interpret the optimality conditions with classical and extended derivatives within the theory of categories.