We study the flat region of stationary points of the functional under the constraint where is a bounded domain of . Here is a function which is concave for small and convex for large, and is a given constant. The problem generalizes the classical minimal resistance body problems considered by Newton. We construct a family of partially flat radial solutions to the associated stationary problem when is a ball. We analyze also some other qualitative properties. Moreover, we show the uniqueness of a radial solution minimizing the above mentioned functional. Finally, we consider nonsymmetric domains and provide sufficient conditions which insure that a stationary solution has a flat part.