In this paper, the polarizability and dipole moment of different types of small inhomogeneous inclusions are solved. From the dipole moment, the effective permittivity of a dielectric mixture that contains these kinds of inclusions are solved. Two ways to approach the exact solution are presented: the external and internal field solutions. For cases where an inclusion does not possess sharp boundaries, a variational approach appears most efficient. A stationary functional for the polarizability is derived which can be used to calculate accurate estimates for the dipole moment with simple trial functions. Results are shown for multilayer inclusions, and inhomogeneous inclusions with linear, parabolic, and Gaussian permittivity profiles.