Eliminating y between (1)' and (3), we get ...(3). terminate form : we may however extricate ourselves from this difficulty by differentiating (2)' successively until we obtain an equation in which no longer presents itself under this dy form. Thus, after one differentiation, dzy dy2 dy (y2 – ax) + 2y dx2 dx2 2a + 2x = 0, Thus we see that the system of values x = ponds to a minimum value of y. We will discuss this example also by examining directly dy whether changes sign as x passes through 0 and a 2. dx First, taking the value 0 of x, put h for x in (1), h being very small: then The values + (3a). h of y must be rejected because they are impossible when - h is put for + h. We will next take the system x = a/2, a 4. Putting x = a/2 + h, y = a 4+ k, in (1)', we have nearly 2a3 + 3a2 (2)3 . h − 3a {a (2)2 + h} . (a 3⁄4/4 + k) whence, approximately, + 4a3 + 3a2 (4)3 . k = 0, 3a2 (2)3. h − 3a {a (4)* h + a (2)3 k} + 3a2 (4)* k = 0, whence hence x = a 2 gives a 4 as a maximum value of y. already considered, and x = a 4, y = a/2, which makes we may shew, in precisely the same kind of way, that these values of the variables do not correspond to a maximum or minimum value of y. Maxima and Minima of a Function of a Function. 76. Suppose that r = f(x), and x = +(0), f(x) denoting a certain function of x, and 4(0) a function of 0; and let it be proposed to determine the maxima and minima of r as depending upon the variation of 0. We know that From this relation it appears that, in order that r may have a maximum or minimum value, the expression ƒ'(x). 4'(0) must experience a change of sign as 0 varies from 0 ̧ – a to 0 ̧ + a, 0, being the value of which makes r a maximum or minimum, and a being an indefinitely small positive quantity. In order that such a change of sign may take place it is necessary that either f'(x) or '(0) change sign, but that both do not change sign at once. In other words, that y may have a maximum or minimum value, it is necessary that either f(x) have a maximum or minimum value as dependent upon the variation of x, or that 4(0) have a maximum or minimum value as dependent upon the variation of 0, and that f(x) and (0) have not maximum or minimum values simultaneously. If f(x) have a maximum or minimum value in relation to x, then r will have respectively a maximum or minimum value also in relation to 0: for, if '(0) be positive, then as 0 varies from 0 ̧ – a to 0 + a, x will vary from xo -h to x + h, and therefore, f'(x) changing sign from + to-, f'(x). (0) will also change sign from + to; and, if '(0) be negative, then as @ varies from 0。 - a to 0。 + a, x will vary from xoh to x - h, and therefore, f'(x) changing sign from - to +, ƒ'(x) . ¥'(0) will change sign from to -: that is, a maximum value of f(x) with regard to a corresponds to a maximum value of r with regard to : we may shew in like manner that a minimum value of f(x) with regard to x corresponds to a minimum value of r with regard to 0. Also, if (0) have a maximum value in relation to 0, r will have a maximum or a minimum value in relation to 0 accordingly as f'(x) is positive or negative; and if (0) have a minimum value in relation to 0, r will have a maximum or a minimum value in relation to 0 accordingly as f'(x) is negative or positive. Ex. 1. Let it be proposed to find the maximum or minimum value of r, when thus changes sign from to as varies from 0 to + 0; dx do and from + to as varies from π if 0 = 0, and if Ꮎ x = 0; r = m, a minimum value of r; = π, x = + ∞ ; r = + ∞, a maximum value of r. This is the solution of the problem " to find the maximum or minimum values of the radius vector of a parabola, the focus being the pole." Ex. 2. To find the maximum or minimum values of an essentially positive quantity, which shews that r has no maximum or minimum in relation to x taken absolutely. through zero, and from + to as 0 increases through 7. Hence This is the solution of the problem "to find the maximum value of the perpendicular drawn from the focus of an ellipse upon the tangent," r being the square of this distance and a the radius vector. For additional examples the reader is referred to a paper in the Cambridge Mathematical Journal, for February, 1843, entitled "On certain cases of Geometrical Maxima and Minima.” Maxima and Minima of a Function of two Independent 77. Let z = f(x, y), x and y being two independent variables. We are at liberty to assume that y = f(x) provided that 4(x) denotes an arbitrary function of x. Differentiating z on this hypothesis, we have Dz dz dz = + y' dx dx dy Now that z may have a maximum or minimum value for any system of values x and yo of x and y, it is sufficient and necessary that, as x increases through o, the total differential Dz Хо Уо coefficient and therefore the expression dx |
