= Solution
For the linear function, $F=-1/2$ and $RF-f=\Lambda$. Therefore
$$
\boxed{V=2\Lambda,\qquad\phi=\sqrt{3/2}\log(1/2).}
$$
The factor two follows from the paper's <conformal transformation> $\widetilde g=g/2$. Its vacuum equation is consequently $\widetilde G_{\mu\nu}=2\Lambda\widetilde g_{\mu\nu}$. Here $f''=0$ and the scalar is fixed: the <Einstein-frame representation of metric f(R) gravity> degenerates to ordinary <general relativity> with a rescaled <cosmological constant>, and supplies no additional propagating scalar field.
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