Design of an optimal extended surface having functionally graded material is significant in cooling performance of hot attached structures in technological applications. The present endeavor is to search for axially variable thermal conductivity formula for a prescribed longitudinal fin shape of rectangular or triangular cross section. Heat transfer is presumed to take place through conductive, convective and radiative effects. The well-known fact is that it is not possible to solve in closed-form the highly nonlinear heat transfer equation under such considerations in general, unless some effects are ignored. Temperature or spatial dependence of material properties of the fin make the problem even harder to treat without numerical simulations. To help designer to avoid such simulations, prescribed temperature distributions in the form of elementary polynomial functions involving some shape parameters are utilized. Under operative geometric and thermal parameters such as the Biot number and the radiation parameter, exact solution formulae for the pertinent thermal conductivity distribution along the functionally graded extended surface are then obtained. The price to pay is only to work out the domain of definition of physical parameters acting on the loaded temperature profile.Designer can benefit from the advantage of the presented elementary solutions while analyzing the efficiency of convecting-radiating longitudinal fins of rectangular, triangular or a more general tapered longitudinal fin class cross sections and control/adjust the physical parameters to the desired temperature/material conditions. With a preloaded temperature profile to the energy equation, the tip temperature can be adjusted so as to enhance the heat transfer rate by increasing/decreasing the governing fin parameters. Such promising inverse problem of extracting axial thermal conductivity distribution from a prescribed temperature solution can also be utilized in other kinds of fin profiles without resorting to the numerical simulations.