We apply an extension of a Leggett-Williams type fixed point theorem to a two-point boundary value problem for a fourth order ordinary differential equation. The fixed point theorem employs concave and convex functionals defined on a cone in a Banach spstce. Inequalities that extend the notion of concavity to fourth order differential inequalities are derived and employed to provide the necessary estimates. Symmetry is employed in the construction of the appropriate Banach space.