FERMAT: ONE VARIABLE WITHOUT CONSTRAINTS 5 1.1 INTRODUCTION Optimization and the differential calculus. The first general method of solution of extremal problems is due to Pierre de Fermat (1608­1665). In 1638 he presented his idea in a letter to the prominent mathematicians Gilles Persone de Roberval (1602­1675) and Marin Mersenne (1588­1648). Scientific journals did not yet exist, and writ- ing a letter to learned correspondents was a usual way to communicate a new discovery. Intuitively, the idea is that the tangent line at the highest or lowest point of a graph of a function is horizontal. Of course, this tangent line is only defined if the graph has no "kink" at this point. The exact meaning became clear later when Isaac Newton (1642/43­ 1727) and Gottfried von Leibniz (1646­1716) invented the elements of classical analysis. One of the motivations for creating analysis was the desire of Newton and Leibniz to find general approaches to the solution of problems of maximum and minimum. This was reflected, in particular, in the title of the first published work devoted to the differential calculus (written by Leibniz, published in 1684). It begins with the words "Nova methodus pro maximis et minimis . . . ."