Several sets of cases of one-dimensional heat conduction in systems with time independent boundary conditions are dealt with to search for the critical or extremum points of entropy production rate and of other functions, with reference also to some papers of Onsager, Li, Prigogine and other authors in the domain of generalized thermodynamics. In heat conduction systems a steady state never corresponds to the state of minimum entropy production. On the other hand, a different function, thermokinetic potential, has always a decreasing trend with time; the condition of minimum thermokinetic potential corresponds exactly to that of the steady state. Besides, if it is impossible to define a thermokinetic potential for a system, then this system may not be stable and a stationary steady state may not exist. The results of these studies may be chiefly utilized for thermodynamic optimization in cryogenic insulation systems. In these systems, indeed, when the thermal conductance has already been decreased to the limit imposed by economic considerations, it is possible to find a temperature distribution, and consequently a thermal flux variation along the insulation, which corresponds to the minimum of the total entropy production rate (in the insulation and in the refrigerating systems that are necessary to maintain a steady state).