  We estimate the displacement energy of Lagrangian 3-spheres in a symplectic
6-manifold $X$, by estimating the displacement energy of a one-parameter family
$L_{\lambda}$ of Lagrangian tori near the sphere. The proof establishes a new
version of Lagrangian Floer theory with cylinder corrections, which is
motivated by the change of open Gromov-Witten invariants under the conifold
transition. We also make observations and computations on the classical Floer
theory by using symplectic sum formula and Welschinger invariants.
