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— ■»■** + Tt —;
vlv) = 2-hx — Ts -,
x2(aa + x*)i
= 0, if x = —, and changes sign from — to +; which indi
cates a minimum; and y = 2a; in which case the semivertical angle = cot-1\/8, and the surface = 2ira*.
Ex. 8. To describe the least cone about a given sphere.
In fig. 18, let the circle Apbp', and the triangle Efg, represent a plane section of the sphere and circumscribed cone, made by the paper passing through the sphere's centre; let the radius of the sphere = a; and Cm = x, Hf = y, so that
x2 + y2 = a2. Then, by properties of the tangent of a circle,
a(a + x) ia + x\b
= 0, and changes sign from -f to —, if x = jr, which indite
cates a maximum; therefore Om= ^, and the area of the greatest parabola = ^- (a2 + 62)*6.
152.] In the preceding Articles we have determined certain properties of maxima and minima, and have investigated methods by which particular cases of such singular values may be found. The general theory however, to be complete, requires a criterion whereby the number of maxima and the number of minima of a given function may be determined; and also means for determining the greatest and the least, or, as I shall call them, the absolute maximum and the absolute minimum, of all the singular values of the function.
Let F(<r) be the function whose singular values are the subject of consideration; and let x0 be the general symbol of the critical value of x to which such singular values correspond; and let y0 be the corresponding value of r(x); and let /3 be the value which the second derived-function of r(,r) takes, when x = x0. Then we have the following equations, when x = x0:
F(aO-y,= 0, (1) ¥\x) = 0, (2) r»-/3 = 0. (3)
Now as the last two equations coexist when relation exists between them which can, theoretically at least, be expressed in terms of /3 and the coefficients of the original function, and which will be independent of x and x0; or, in other words, x may be eliminated from (2) and (3). If this relation is found, and is arranged in terms of /3, we shall have an equation in terms of /3, the signs of the roots of which will enable us to distinguish maxima and minima; because there will be as many maxima values as there are negative values of /3, and as many minima values as there are positive values. In many cases however this process will not enable us to determine the number of the singular values, because there may be many of the same value. Thus if r(x) = sinx, r'(x) = cos<r, F"(.r) =
— sin x; and we have to eliminate x between cos x = 0, and
— sin x —fi = 0; whence we have /3J —1 =0, and /3 = 1, and /3 = — 1; hence sin x has one minimum value and one maximum value, but it has an infinite number of maxima and minima, corresponding to the infinite number of the roots of the equation f\x) = 0, or cos x = 0.
The determination of the required equation may be performed in any way that is practicable, although it is in most cases beyond our power; if however F(x) is a rational and integral function of x, then, because (2) and (3) are satisfied when x = x0, they have a common measure of the form (x—x0)m, where m is either unity or some positive integer greater than unity; and if we proceed by the ordinary algebraical process of finding this common measure, we shall finally come to a remainder independent of x and in terms of /3, which must be equal to zero; and if the terms of it are arranged in powers of /3, we shall have an equation the signs of the roots of which will fix the numbers of the maxima and the minima of the original function. The following examples illustrate the process.
Ex.1. Let ¥(x) = x3-2xi + x;
.-. r'(x) = 3^ — 4^-1-1,
If we proceed by the ordinary method to find the greatest common measure between 3#2 — 4^ + 1 and 3j,—2—/3, we shall after two divisions come to the remainder /32 —1, which must be equal to zero; whence we have /3 = +1; and as the values of j8 are one positive and one negative, so the singular values of r(x) are respectively a minimum and a maximum.
Ex. 2. F(x) = xs + axi-2aix,
F'(ar) = 3.r2 + 2a.r-2as,
If we proceed as in the last example we shall find ft2 —7 a2 = 0; so that ft — +7* a; and therefore F (,r) has two singular values which are respectively a maximum and a minimum.
If one of the values of ft, deduced from the result of the elimination between (2) and (3), is zero, the criterion fails for a singular value, or for the particular kind of it, corresponding to that value of the variable which makes ft = 0. Of this circumstance the following is an example:
r(x) = 3x*-l6xs + 24x2~\\,
v'(x) = 12(^-4^ + 4^),
F'v) = 12(3*»-8aM 4). Now in pursuit of the process by which, as in the preceding examples, the criteria of maxima and minima have been determined, let us endeavour to find the common measure of aP—iaP + ix and of 3x2 — Sx + 4 — ft; then the first remainder which does not involve x is 3ft2 — 32/3; and of this when equated to zero the roots are 0 and a positive quantity; so that corresponding to the factors which make to vanish, for one r(x) may have no singular value; and the other gives to F(x) a minimum value, viz. —11.
153.] Let thus much suffice for the number of maxima and the number of minima of a given function r(x); and let us now investigate a method for determining the greatest and least absolutely of all the singular values which v(x) has. Since (1) and (2) of the last Article are both satisfied when x = x0, they have a common measure. When T(x) is transcendental, the determination of this relation is generally beyond our powers; but when F(x) is a rational and integral algebraical function, it can be found. For if we operate on (1) and (2) in the ordinary process of finding the greatest common measure, we shall finally arrive at a remainder which is independent of x, but which involves y0 and the coefficients of the original function; this remainder must be equal to zero; and thus we shall have
Trick, Vol. I. K k
an equation in terms of y0, the roots of which will be the singular values of F (x); and thus the greatest root will be that value of r(x) which is absolutely the greatest, and the least root will be the absolute minimum. The process is illustrated by the following examples.
Ex.1. r(x) = xs-2x* + x,
r'(x) = 3x2—4# + l.
If, as in Art. 152, y0 is the singular value of r(x), and if we pursue the common process for the determination of the common measure of xa — 2x2 + x —y0 and of 3x*—4x +1, the first remainder which is independent of J? is 27 y02 —4yo; and this is to be equal to zero; so that the two singular values of y0 are 4
0 and of which the former is the absolute minimum, and
the latter is the absolute maximum of all the relative maxima and minima.
Ex.2. K(^) = ^-40^-80*^-)-11a4,
r'(x) = 4^-120^-160^.
If we proceed to find the greatest common measure of r(x) — y0, and of v'(x), the first remainder which is independent of x is yQs + 109o4y0a — 936 a9 y0, of which, when equated to zero, the roots are 0, 8 a*, —117 o*: whence we know the relative maxima and minima, and therefore the absolute values also.
Section 2.—On maxima and minima of implicit functions of two variables.
154.3 Suppose that it is required to find the values of x which make y a maximum or minimum, when the equation connecting y and x is of the form, u = F (x, y) = c; then, by Art. 48,
dy = (4) dz idu\'
and as the necessary condition of such a singular value of y is, dv
that ~ changes sign, and as a change of sign can take place
only by being equal to 0 or to 00, it follows that either