  In this paper we consider the second eigenfunction of the Laplacian with
Dirichlet boundary conditions in convex domains. If the domain has \emph{large
eccentricity} then the eigenfunction has \emph{exactly} two nondegenerate
critical points (of course they are one maximum and one minimum). The proof
uses some estimates proved by Jerison ([Jer95a]) and Grieser-Jerison ([GJ96])
jointly with a topological degree argument. Analogous results for higher order
eigenfunctions are proved in rectangular-like domains considered in [GJ09].
