The consequences of using the idea of entropy production rate minimization are examined in relation to the shape of radial temperature profiles and the scaling of transport coefficients with temperature in Tokamak discharges. This is done under the assumption that physical processes occur in such a way as to produce relaxed states where an extremal of some relevant quantity (in this case the entropy production) is obtained. The authors study the conditions under which the temperature profiles adopt the Gaussian-like shapes observed in Tokamak plasmas. This is possible only for certain scalings of the thermal conductivity with temperature favouring certain models of anomalous transport against others. These results are also dependent on whether or not the current density profile is assumed to be determined by the temperature profile, as in Ohmically heated discharges with constant electric field. In the former case an anomalous electrical resistivity is needed in order to obtain the correct temperature profiles for known models of anomalous thermal conductivity.