  We study the symplectic geometry of derived intersections of Lagrangian
morphisms. In particular, we show that for a functional $f : X \rightarrow
\mathbb{A}_k^1$, the derived critical locus has a natural Lagrangian fibration
$\textbf{Crit}(f) \rightarrow X$. In the case where $f$ is non-degenerate and
the strict critical locus is smooth, we show that the Lagrangian fibration on
the derived critical locus is determined by the Hessian quadratic form.
