MAXIMA AND MINIMA, in mathematics. By the maxi mum or minimum value of an expression or quantity is meant primarily the "greatest" or "least" value that it can receive. In general, however, there are points at which its value ceases to in crease and begins to decrease ; its value at such a point is called a maximum. So there are points at which its value ceases to de crease and begins to increase; such a value is called a minimum. There may be several such maxima or minima, and such a mini mum is not necessarily less than such a maximum. For instance, the expression I) can take all values from 00 to I and from +7 to +00 but has, so long as x is real, no value between I and +7. Here I is a maximum value, and +7 is a minimum value of the expression, though it can be made greater or less than any assignable quantity.
The first general method of investigating maxima and minima seems to have been published in A.D. 1629 by Pierre Fermat. Particular cases had been discussed. Thus Euclid in book III. of the Elements finds the greatest and least straight lines that can be drawn from a point to the circumference of a circle, and in book VI. (in a proposition generally omitted from editions of his works) finds the parallelogram of greatest area with a given perim eter. Apollonius investigated the greatest and least distances of a point from the perimeter of a conic section, and discovered them to be the normals, and that their feet were the intersections of the conic with a rectangular hyperbola. Some remarkable theorems on maximum areas are attributed to Zenodorus, and preserved by Pappus and Theon of Alexandria. The most noteworthy of them are the following: I. Of polygons of n sides with a given perimeter the regular polygon encloses the greatest area.
2. Of two regular polygons of the same perimeter, that with the greater number of sides encloses the greater area.
3. The circle encloses a greater area than any polygon of the same perimeter.
4. The sum of the areas of two isosceles triangles on given bases, the sum of whose perimeters is given, is greatest when the triangles are similar.
5. Of segments of a circle of given perimeter, the semicircle encloses the greatest area.
6. The sphere is the surface of given area which encloses the greatest volume.
The next problem on maxima and minima of which there ap pears to be any record occurs in a letter from Regiomontanus to Roder (July 4, 1471), and is a particular numerical example of the problem of finding the point on a given straight line at which two given points subtend a maximum angle. Tartaglia in his Gen eral trattato de numeri et mesuri (c. 1556) gives, without proof, a rule for dividing a number into two parts such that the contin ued product of the numbers and their difference is a maximum.
Fermat investigated maxima and minima by means of the prin ciple that in the neighbourhood of a maximum or minimum the differences of the values of a function are insensible, a method virtually the same as that of the differential calculus, and of great use in dealing with geometrical maxima and minima. His method was developed by Huygens, Leibniz, Newton and others, and in particular by John Hudde, who investigated maxima and minima of functions of more than one independent variable, and made some attempt to discriminate between maxima and minima, a question first definitely settled, so far as one variable is concerned, by Colin Maclaurin in his Treatise on Fluxion (1742). The
method of the differential calculus was perfected by Euler and Lagrange.
Jean (Johann) Bernoulli's famous problem of the "brachisto chrone," or curve of quickest descent from one point to another under the action of gravity, proposed in 1696, gave rise to a new kind of maximum and minimum problem in which we have to find a curve and not points on a given curve. From these problems arose the "Calculus of Variations." (See CALCULUS OF VARIA TIONS.) The method of the differential calculus is theoretically very simple. Let u be a function of several independent variables xl, xn, if u is a maximum or minimum for the set of values . . and u becomes u+ou, when x3, . . . receive small increments . . . (1;6 then Su must have the same sign for all possible values of . . The sign of this expression in general is that of , which cannot be one-signed when (VI, 6x2, ox. can take all possible values, for a set of increments . . ox will give an opposite sign to the set . . . --ox. Hence must vanish for all sets of increments . . . 6x, and since these are independent, we must have =o, . . A value of u given by a set of solu tions of these equations is called a "critical value" of u. The value of Su now becomes for u to be a maximum or minimum this must have always the same sign. For the case of a single variable x, corresponding to a value of x given by the equation du/dx=o, u is a maximum or minimum as is negative or positive. If vanishes, then there is no maximum or minimum unless vanishes, and there is a maximum or minimum according as is negative or positive. Generally, if the first differential coefficient which does not vanish is even, there is a maximum or minimum according as this is negative or positive. If it is odd, there is no maximum or minimum.
In the case of several variables, the quadratic for a maximum or minimum at all and and both negative for a maximum, and both positive for a minimum. It is important to notice that by the quadratic being one-signed is meant that it cannot be made to vanish except when Sx1, . . . all vanish. If, in the case of two variables, then the quadratic is one-signed unless it vanishes, but the value of u is not necessarily a maximum or minimum, and the terms of the third and possibly fourth order must be taken into account. A critical value usually gives a maximum or minimum in the case of a function of one variable, and often in the case of several independent variables, but such maxima and minima are purely local and the absolutely greatest and least values are not neces sarily critical values. If, for example, x is restricted to lie be tween the values a and b and 4'(x) =o has no roots in this in terval, it follows that (15/(x) is one-signed as x increases from a to b, so that 0(x) is increasing or diminishing all the time, and the greatest and least values of 4(x) are 4(a) and 4(b), though neither of them is a critical value. In general, the absolutely greatest and least values of the function may be given by Ca) or c/)(b), however many critical values exist.
Full analytical details may be found in any standard treatise on the Calculus. English writers, however, are apt to ignore any but critical values. See MATHEMATICAL MODELS. (A. E. J.)