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 nondegenerate 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.