  In the framework of classical scale invariance, we consider quadratic gravity
in the Palatini formalism and investigate the inflationary predictions of the
theory. Our model corresponds to a two-field scalar-tensor theory, that
involves the Higgs field and an extra scalar field stemming from a gauge
$U(1)_X$ extension of the Standard Model, which contains an extra gauge boson
and three right-handed neutrinos. Both scalar fields couple nonminimally to
gravity and induce the Planck scale dynamically, once they develop vacuum
expectation values. By means of the Gildener-Weinberg approach, we describe the
inflationary dynamics in terms of a single scalar degree of freedom along the
flat direction of the tree-level potential. The one-loop effective potential in
the Einstein frame exhibits plateaus on both sides of the minimum and thus the
model can accommodate both small and large field inflation. The inflationary
predictions of the model are found to comply with the latest bounds set by the
Planck collaboration for a wide range of parameters and the effect of the
quadratic in curvature terms is to reduce the value of the tensor-to-scalar
ratio.
