In this paper, the conditions of the existence of backward-wave modes in closed, lossless waveguides filled with inhomogeneous and anisotropic medium that has coupling between transverse and longitudinal field components are presented. In these waveguides, Maxwell's equations are transformed into an infinite linear algebraic equation system by application of the Galerkin version of moment method. Propagation constants of the problem are found as the eigenvalues of the coefficient matrix of this infinite linear algebraic equation system. In this paper, the derivative of the eigenvalue is obtained analytically using the result expressions of moment method and group velocity is determined. It is utilized to reveal necessary and sufficient conditions for the existence of backward-wave mode. These conditions are adequate to determine whether this waveguide supports the backward-wave mode in a frequency range of interest.