and that of the derived functions ƒ"(x), ƒ"(x),.... the first which, for a corresponding value of x, does not vanish, shall be of an even order and shall be positive. Hence, to find the maxima and minima of a function f(x), we must equate its first differential coefficient to zero, and thence obtain corresponding values of x: we must then keep differentiating the function until, for each of these values of x, we arrive at a differential coefficient which does not vanish: if, for any one of these values of x, this final differential coefficient is of an even order, the corresponding value of f(x) will be a maximum or a minimum accordingly as the final differential coefficient is negative or positive. If the final differential coefficient is of an odd order, the corresponding value of f(x) will be neither a maximum nor a minimum. (x) being an essentially positive factor. Take then, instead the roots of this equation are 1, 2, 3. Now 12x2 48x + 44: 1 makes it a Thus we see that for the values 1, 2, 3, of x, y is respectively a minimum, a maximum, a minimum, where (x) = (x − a), a quantity essentially positive. Instead, therefore, of dy or f'(x), we may take dx $(x) = = c(x − a) = 0 ; '(x) = C. whence we get x = a; also a minimum ; If therefore c be a positive quantity, x = a makes y From this and the preceding example it appears that, although either f(x), or its derived functions of sufficiently high orders, may become infinite for a value of x which makes f(x) a maximum or a minimum, yet, if we replace f'(x) by an appropriate function (x) which has always the same sign as ƒ'(x), we may often apply with advantage the rule of Art. (73) for finding such a value of x. Abbreviation of Operation. 74. Suppose that, for a certain function f(x) = y, u being a factor which vanishes when x = a, while v remains finite. Then, if d'u be the first differential coefficient of u which does not vanish when x = a, it is easy to see that and that all the differential coefficients of y of lower orders than the 7th will be equal to zero. This consideration enables us to ascertain whether a particular value a of x, which makes dy dx = 0, corresponds to a maximum or a minimum value of y, without being driven to the necessity of obtaining the general expression Ex. 1. Suppose that y is such a function of x that and suppose that we desire to know whether x = dy dx 1, which makes = 0, corresponds to a maximum or to a minimum value of y. We have, if x = 1, d'y dx2 = (x − 2) (x − 8) (x − 4) = (-) (-) (-) = -: which shews that = 1 corresponds to a maximum value of y. which shews that x = 1 corresponds to a maximum value of y. Maxima and Minima of implicit Functions of a single Variable. 75. In the preceding articles we have investigated the method of finding the maxima and minima of an explicit function of a single variable. We proceed now to the consideration of those cases in which the function is involved implicitly with its variable. Let y be a function of xæ by virtue of an equation 0 or = ∞ is an essential condition for a From (1) and (3) we may obtain those systems of values of x and y which alone can correspond to maximum or minimum values of y. In order to find whether a value x of x so discovered does really give a maximum or minimum value of y, we must substitute, in the expression - h, xh, successively for x, and the corresponding values of y, and see whether the expression changes sign from + to F, the additional conditions for a maximum and a minimum value respectively. For those values of x which do not render dy dx or any of the higher order of differential coefficients of y, equal to infinity, we may proceed to find, by implicit differentiation, the values of d'y d3y until we arrive at one which does not vanish for dx2' dx3 ... a value of x which makes dy dx = 0. If the order of this final differential coefficient be even, the value of x gives a maximum or a minimum value of y, accordingly as the sign of the differential coefficient is negative or positive. Ex. Let x3 - 3a xy + y3 = 0. . . . . .(1)': |
