Summary: In this paper we study the classification of the \( f\)-minimal hypersurface immersed in the manifold \( M^{n}\times\mathbb R\), where \( (M^{n}, g)\) is an Einstein manifold with positive Ricci curvature. By using the \( F\) functional and \( F\)-stability which were introduced by Huisken and Colding-Minicozzi respectively, we prove that among all complete \( f\)-minimal hypersurfaces with polynomial volume growth, only \( M^{n}\times \{0\}\) is \( F\)-stable.