CHAPTER VII. MAXIMA AND MINIMA. SECT. 1. Explicit Functions of One Variable. SUPPOSE that u is any explicit function of a: the following rule will enable us to determine those values of a which du render u a maximum or minimum. "Equate to zero or dx infinity let a be a possible value of a obtained from either du of these equations; then, if changes sign from + to or from to when, h being an indefinitely small quantity, a-h and a +h, are substituted successively for x, x = a will correspond respectively to a maximum or minimum value of u if no such change of sign takes place the value a of a must be rejected. By applying this process to each of the : of a essentially positive for all possible values of x: then, du instead of we may evidently take v = (a), and treat v dx The following principle is also frequently useful for the determination of maxima and minima. "Suppose that, for any particular value of x, du da = = 0, and that du d'u ď3u dx' da da3 are none of them infinite: then, if the first of these differential coefficients which does not vanish, for the particular value of a, be of an even order, u will be a maximum or a minimum accordingly as this differential coefficient is negative or positive." If du dx = †(x).v, †(x) being an es sentially positive function of a, the following modification of this principle in many cases affords considerable simplification. "Suppose that, for any particular value of x, v = 0, and that dv dev d3 v dx' dx2' dx3 are none of them infinite: then, if the first of these differential coefficients which does not vanish, for the particular value of x, be of an odd order, u will be a maximum or a minimum accordingly as this derived function is negative or positive." efficient of u which does not vanish for a particular value a of a, whether the value of u be a maximum or a minimum, the following consideration will sometimes shorten the process. value of x, which causes one of the factors as w, and its first n2 differential coefficients to vanish, the dru which is to be considered is that involving only term of as dan-1 dr u dx is reduced to one term. dx" The investigation of the maximum and minimum values of u is sometimes facilitated by the following considerations. If u be a maximum or minimum, and a be a positive constant, au is also a maximum or minimum. but When u is a maximum or minimum, au2n+1 is so also ; a 2n+1 is inversely a minimum or maximum. 2n If u be a positive maximum or minimum, au2" is also a maximum or minimum. If u be a negative maximum or minimum, au2" will be a minimum or maximum. The same remarks apply to fractional powers of the function u, except that when the denominator of the fraction is even, and the value of u negative, the power of u is impossible. When u is a positive maximum or minimum, logu is a maximum or minimum. This preparation of the function is frequently made when the function u consists of products or quotients of roots and powers, as the differentiation is thus facilitated. Other transformations of u are sometimes useful, but as these depend on particular forms which but rarely occur, they may be left to the ingenuity of the student who desires to simplify the solution of the proposed problem. 2, and x = makes u, a maximum. = u = x1 — 8 x3 + 22x2 - 24x + 12. The roots of this equation are 1, 2, 3, and The roots o make u neither a maximum nor a minimum; nor a minimum when n is odd, because x = a makes u = 0, which is a minimum when n is even, because du da are substituted successively for x; and is then insusceptible to when a changes sign from neither a maximum du dx the roots of which are a = 0, x = a, and ≈ = ma m + n x = 0 makes u = 0, a minimum if m be even, and neither a maximum nor a minimum if m be odd. x = a makes u = 0, a minimum if n be even, and neither a maximum nor a minimum if n be odd. d'u d − − dx2= √ {a1-1 (a−x)" -1} {ma−(m+n)x} − (m+n)x"−'(a−x)"−1, dx |