  We study the Du Bois complex $\underline{\Omega}_Z^\bullet$ of a hypersurface
$Z$ in a smooth complex algebraic variety in terms its minimal exponent
$\widetilde{\alpha}(Z)$. The latter is an invariant of singularities, defined
as the negative of the greatest root of the reduced Bernstein-Sato polynomial
of $Z$, and refining the log canonical threshold. We show that if
$\widetilde{\alpha}(Z)\geq p+1$, then the canonical morphism $\Omega_Z^p\to
\underline{\Omega}_Z^p$ is an isomorphism, where $\underline{\Omega}_Z^p$ is
the $p$-th associated graded piece of the Du Bois complex with respect to the
Hodge filtration. On the other hand, if $Z$ is singular and
$\widetilde{\alpha}(Z)>p\geq 2$, we obtain non-vanishing results for some of
the higher cohomologies of $\underline{\Omega}_Z^{n-p}$.
