The problem of optimization of the first resonant frequency of a rod made of a functionally graded material (FGM) is considered, where the variable elastic modulus is the only control function. Two types of constraints are imposed on the laws of its variation: an isoperimetric constraint on the mean value and a constraint on the minimum value. Previously similar problems without constraints of the second type were studied. In this case, optimal solutions for some boundary conditions led to piecewise smooth laws that vanish inside the region occupied by the rod. The second type of constraint is introduced so that the function characterizing the lawof variation of Young’s modulus doesn’t vanish inside the rod. The geometric characteristics of the rod, such as thickness, moment of inertia, and cross-sectional area are assumed to be constant. Note that modern production of FGM and structural elements made of them allows for the variability of elastic properties in fairly wide limits, but still not as wide as the geometric dimensions of the beam could change. Therefore, the main objective of the work is to construct an optimal solution that ensures the maximum of the first natural frequency and satisfies the imposed constraints. Using the Rayleigh variational approach, the optimality condition is obtained and the existence of homogeneous and heterogeneous regions of variation of Young’s modulus for the optimal design is substantiated. Optimal solutions are constructed for the problem of bending vibrations for three types of boundary conditions: cantilever clamping, clamping of both ends, and spring-type boundary conditions. The optimality condition allows one to write the general form of the solution in heterogeneous regions as a polynomial: of degree 2 for the deflection function and degree 4 for Young’s modulus. In homogeneous regions, the solution has a standard form for a beam and is expressed through trigonometric and hyperbolic functions. The problem of conjugating solutions leads to systems of linear algebraic equations, which are also solved analytically. Only the frequency parameter and the parameters characterizing the transition points between homogeneous and heterogeneous regions are defined numerically, since its enter into the obtained relations in a substantially nonlinear manner. The dependence of the fundamental frequency of oscillations and the shapes of regions where the material is homogeneous on the parameter characterizing thelower limit for Young’s modulus is analyzed.