  We prove a new rigidity result for metrics defined on closed smooth $ n
$-manifolds that are critical for the quadratic functional $ \mathfrak{F}_{t}
$, which depends on the Ricci curvature $ Ric $ and the scalar curvature $ R $,
and that satisfy a pinching condition of the form $ Sec > \epsilon R $, where $
\epsilon $ is a function of $ t $ and $ n $, while $ Sec $ denotes the
sectional curvature. In particular, we show that Bach-flat metrics with
constant scalar curvature satisfying $ Sec > \frac{1}{48} R $ are Einstein and,
by a known result, are isometric to $ \mathbb{S}^{4} $, $ \mathbb{RP}^{4} $ or
$ \mathbb{CP}^{2} $.
