ABSTRACT
The concept of a convex set is used to define concave and convex functions. Although the names appear similar, a convex set should not be confused with a convex function. However, a concave and a convex function are defined in terms of a convex set. Geometrically, convex function is convex if its epigraph is a convex set. An epigraph is convex if every hyperplane intersecting it produces a convex shaped slice. This theorem is about a convex slice which is cut out of the epigraph by a vertical hyperplane. The first-order direct partial derivatives must be zero simultaneously, which means that at a critical point the function is neither increasing nor decreasing with respect to the principal axes. The second-order partial derivatives, when calculated at the critical point, must be negative for a relative maximum and positive for a relative minimum.
