In this paper we consider optimal distribution of a gradient material at the circular cross section of a cantilever. The design variables are the parameters which express the distribution of the material density or the elastic constant, and Young's modulus is assumed to be linear in the material density. Two problems are analyzed : (1) maximization of the bending stiffness with constant weight, and (2) minimization of the maximum stress at the cross section with constant bending stiffness. In a general case, the above problems reduce to generalized eigenvalue problems which are formulated and solved by symbolic manipulation and numerical calculation. The optimal distribution indicates that for (1), the density is concentrated at the circular surface such as of a circular cylinder, and for (2), Young's modulus decreases with respect to the radius. The obtained results are also applicable to the optimal design of composite materials where the distribution of the material density is not continuous.