At a constrained optimum (maximize $f$ along the curve $g = c$), Lagrange says $\nabla f = \lambda\nabla g$. What is the geometric picture behind this equation?

C. The level curve of $f$ is tangent to the constraint

While level curves cross the constraint, sliding along it still improves $f$; the climb halts exactly where a level curve kisses the constraint tangentially. Parallel tangents mean parallel normals: $\nabla f = \lambda\nabla g$, with $\lambda$ measuring how hard the constraint is holding you back.