On Extrema III

"In order to regain in a rigorously defined function those properties that are analogous to those ascribed to an empirical curve with respect to slope and curvature (first and higher difference quotients), we need not only to require that the function is continuous and has a finite number of maxima and minima in a finite interval, but also assume explicitly that it has the first and a series of higher derivatives (as many as one will want to use)." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"A variety of natural phenomena exhibit what is called the minimum principle. The principle is displayed where the amount of energy expended in performing a given action is the least required for its execution, where the path of a particle or wave in moving from one point to another is the shortest possible, where a motion is completed in the shortest possible time, and so on." (James R Newman, "The World of Mathematics" Vol. II, 1956)

"In the mathematical theory of the maximum and minimum problems in calculus of variations, different methods are employed. The old classical method consists in finding criteria -as to whether or not for a given curve the corresponding number assumes a maximum or minimum. In order to find such criteria a considered curve is a little varied, and it is from this method that the name 'calculus of variations' for the whole branch of mathematics is derived." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"We frequently find that nature acts in such a way as to minimize certain magnitudes. The soap film will take the shape of a surface of smallest area. Light always follows the shortest path, that is, the straight line, and, even when reflected or broken, follows a path which takes a minimum of time. In mechanical systems we find that the movements actually take place in a form which requires less effort in a certain sense than any other possible movement would use. There was a period, about 150 years ago, when physicists believed that the whole of physics might be deduced from certain minimizing principles, subject to calculus of variations, and these principles were interpreted as tendencies--so to say, economical tendencies of nature. Nature seems to follow the tendency of economizing certain magnitudes, of obtaining maximum effects with given means, or to spend minimal means for given effects." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"While the minimum and maximum problems of calculus of variations correspond to the problem in the ordinary calculus of finding peaks and pits of a surface, the minimax problems correspond to the problem of finding the saddle points of the surface (the passes of a mountain)."(Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"Extrema is the generic term for the concepts 'maximum' and 'minimum' , like 'parents' is the generic term for 'father' and 'mother'. Extremal problems have to do with finding maxima and minima. We encounter them everywhere. It is hardly an exaggeration to say that all problems solved by living organisms are those involving a search for extrema." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Maximum and minimum always exist together: if our cup-like surface is turned over, we get a cap, in which the highest point (maximum) corresponds to the lowest point of the cup (minimum). By climbing to the uppermost peak of a mountain we can find ourselves (via reflection in a nearby lake) in the lowest point of the valley below. Here, the mathematician calmly reasons to within an accuracy that amounts to the opposite, so to say, for if we find a maximum and then view the situation from another angle, we see a minimum. The answer thus depends solely on how we view the surface. That is why we always speak of seeking an extremum and not, separately, a maximum or a minimum." (Yakov Khurgin, "Did You Say Mathematics?", 1974) 

"A proven theorem of game theory states that every game with complete information possesses a saddle point and therefore a solution." (Richard A Epstein, "The Theory of Gambling and Statistical Logic" [Revised Edition], 1977)

"So the strategy of mixing the choices with equal likelihood is an equilibrium point for the game, in the same sense that the minimax point is an equilibrium for a game having a saddle point. Thus, using a strategy that randomizes their choices, Max and Min can each announce his or her strategy to the other without the opponent being able to exploit this information to get a larger average payoff for himself or herself." (John L Casti, "Five Golden Rules", 1995)

"What's important about a saddle point is that it represents a decision by the two players that neither can improve upon by unilaterally departing from it. In short, either player can announce such a choice in advance to the other player and suffer no penalty by doing so. Consequently, the best choice for each player is at the saddle point, which is called a 'solution' to the game in pure strategies. This is because regardless of the number of times the game is played, the optimal choice for each player is to always take his or her saddle-point decision. […] the saddle point is at the same time the highest point on the payoff surface in one direction and the lowest in the other direction. Put in algebraic terms using the payoff matrix, the saddle point is where the largest of the row minima coincides with the smallest of the column maxima." (John L Casti, "Five Golden Rules", 1995)