Scalar-field potential for viable models in $f(R)$ theory

The $f(R)$ theory of gravity can be expressed as a scalar tensor theory with a scalar degree of freedom $\phi$. By a conformal transformation, the action and its Gibbons-York-Hawking boundary term are written in the Einstein frame and the field equations are found. An effective potential is defined from part of the trace of the field equations in such a way that it can be calculated as an integral of a purely geometric term. This potential as well as the scalar potential are found, plotted and analyzed for some viable models of $f(R)$ and for two other proposed new, shown viable, models.