  We prove that the derived critical locus of a $G$-invariant function
$S:X\to\mathbb{A}^1$ carries a shifted moment map, and that its derived
symplectic reduction is the derived critical locus of the induced function
$S_{red}:X/G\to\mathbb{A}^1$ on the orbit stack. We also provide a relative
version of this result, and show that derived symplectic reduction commutes
with derived lagrangian intersections.
