  In this work we present some cosmologically relevant solutions using the
spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime in metric
$f(R)$ gravity where the form of the gravitational Lagrangian is given by
$\frac{1}{\alpha}e^{\alpha R}$. In the low curvature limit this theory reduces
to ordinary Einstein-Hilbert Lagrangian together with a cosmological constant
term. Precisely because of this cosmological constant term this theory of
gravity is able to support nonsingular bouncing solutions in both matter and
vacuum background. Since for this theory of gravity $f^{\prime}$ and
$f^{\prime\prime}$ is always positive, this is free of both ghost instability
and tachyonic instability. Moreover, because of the existence of the
cosmological constant term, this gravity theory also admits a de-Sitter
solution. Lastly we hint towards the possibility of a new type of cosmological
solution that is possible only in higher derivative theories of gravity like
this one.
