Inverse extreme problems for a stationary system of thermal mass transfer equations describing the propagation of a substance in a viscous incompressible thermal conductive fluid in a limited region with a lipstick boundary are considered. These tasks are to find unknown parameters of the medium or densities of sources for certain information about the solution. The solvability of direct edge and reverse extreme problems is investigated, the application of the Lagrange principle is justified, optimality systems are derived and analyzed, sufficient conditions for the uniqueness of solutions are established.