MAXIMA AND MINIMA (Lat., neut. pl. of maximns. greatest, and minimus, least). In mathematics, the greatest and the least values of variable quantities or mag,nitudes. Strictly, a maximum is not necessarily the greatest of all the possible values of a variable; it is a value that is greater than the values immediately pre ceding and following it in series. Similarly, a minimum, strictly defined, is a value that is less than the values immediately preceding and fol lowing it. Renee a function may have several maxima and minima, equal or unequal among tilemsel yes. Thus, in the accompanying figure, are maximum values of time ordinates of f (x), and b., are minimum values. The tangent of the angle which a line tangent at any point to the curve makes with time X-axis is zero at a maximum or minimum value of the ordin ate. This means that the differential coedicient dy = 0 (see CALCULUS ), and hence the abseis sas corresponding to the maxima and minima are // the roots of = 0.
dx A function of two variables, y). has a maximum value when fix. n) > f 4- h. y 4- k), for all small values of it and k, positive or ne.ative; and a minimum vo Inc when f ix, y) ix h, p k). The condi tions for maxima and minima in the case of a ari fitnetion n of two variables are 8.r — = 0, and Su — = 0. If A= -,II — and C= 84 &ray the further conditions for a maXinnim are < AC and A < 0, and for a minimum B' < AC and A > O. When B' = AC or A = B
= C = 0, further investigation is necessary.
A few of the important propositions of plane maxima and minima are: (1) Of all triangles formed with the same two given sides, that is the maximum whose sides contain a right angle; (2) of all isoperimetric triangles (those of equal perimeters) on the, same base, the isosceles is the maximum; (3) of all isoperimetric triangles, that which is equilateral is the maximum; (4) of all triangles having the same base and area, the isosceles has the minimum perimeter; (5) if a line of given length be bent and its ends joined by a straight line, the area of the figure inclosed is a nmximum when the curved line has the form of a semicircle; (6) of all isoperi metric plane figures, the maximum is a circle; (7) of all isoperimetric polygons of a given num ber of sides, the maximum is regular.
Traces of the doctrine of maxima and minima are to be found in the works of Apollonius on conic sections, and among the theorems of Zeno dorus. The Hindus displayed great ingenuity in solving, by ordinary algebra, problems of maxima and minima ; but thorough investigation of the subject requires the aid of the calculus, and Kepler, the Bernoulli brothers, Newton, Maclaurin, Euler, and Lagrange distinguished themselves in this department. See CALCULUS.