262 5.1 INTRODUCTION CHAPTER 5 The material in this chapter is mainly of theoretical interest: to solve a concrete optimization problem, one only needs the first order conditions (Fermat, Lagrange, Karush-Kuhn-Tucker, John) together with the Weierstrass theorem. It gives insight into the gap between stationarity and local optimality and shows how this gap can be filled. The words of Liouville from the epigraph can be applied to any situ- ation where you have not yet reached the essence of things (Liouville expressed his doubts on a "proof" of Fermat's Last Theorem that was presented at a meeting of the French Academy). The first order conditions given so far are not the only obstructions to local optimality in smooth problems. For example, these conditions do not distinguish maxima from minima. Therefore, we complement the first order conditions with second order conditions. We will see that there are essentially no other obstructions to local optimality. To be more precise, we will see that the first order conditions together with a slight strengthening of the second order conditions are sufficient for local optimality. Thus, second order conditions often allow us to verify that a stationary point is a local minimum or maximum.