23 ?>ut we have already seen that h 2 f(a + 70 -/(a) = hf (a) + /"(a + 0/0 , f(a - A) -f(a] = - 7y + ^ f"(a - 6h) , where is >0 and <1. Now, when f (a) is finite, it is plain that f(a + h)-f(a) and f(a - h) -f(a) cannot have both the same sign, when h is very small, unless f(a) = Q. Accordingly, the roots of the equation / (x)==0 furnish in general the values of a; for which /(x) has a maximum or a minimum value. Also we have in this case f(a + 70 -/() = j-^ /"( + eh) , f(a - 70 -f(a) = j-1 f"(a - eh) . Consequently, when /"(a) is negative, the corresponding value of f(a) is a maximum; and when f"(a) is positive, f (a) is a minimum. If, however, f (a) vanishes, along with f(a), it is readily seen that the corresponding value of u is neither a maximum nor a minimum unless / "(a) also vanish. In general, let / ( "H) be the first derived function that does not vanish ; then, if n be odd, the corresponding value of u is neither a maximum nor a minimum ; but if n be even, the corresponding value is a maximum when f^"d) is negative and a minimum when it is positive. These rules for distinguishing between maxima and minima were first given correctly by Maclaurin, in his Fluxions, ch. ix. If the equation f (x) = has no real solution, then/(x) has no maximum or minimum value, and consequently is capable of having all possible values from + co to - o . We shall illustrate the preceding theory by applying it to a few simple cases. du . a Here = 2x + a=0, . . x= . dx 2 Again . 1 ~ = 2. Since * his is a positive quantity, the function is a minimum when x = -~- . Its minimum value is b ; as is also evident because u (2) Here ay? -f 2bx + c u *=* . a x 2 + 2b x + c + llx + c = u(a x 2 + Ib x + c ) . du Differentiate both sides, and, since = for a maximum or a dx minimum, we have ax + 1 = (a x + b )u , Hence the roots of the quadratic (aV - ba ^x 2 + (ac - ca )x + be - cV = give the required solutions. The corresponding values of u are given by the quadratic u*(b 2 - a c") + u(ac f + ca - 2bb ) + b*-ac=Q. If the roots of the quadratic in x be imaginary, the proposed fraction has no maximum or minimum value. When the roots are real, the fraction has one maximum and one minimum value. These can be easily distinguished in any particular case. It is easily seen that to the greater root corresponds a minimum, and to the lesser a maximum value of the fraction, in general. (3) w = tan x x. Here du ., -T- = sec^x - 1 , d 3 u
jT =,2 sec-a; tan x , -7-5=2 sec 4 x - 4 sec 2 x tan 2 x . Hence, for a maximum or a minimum we have sec 2 = 1, . . tan x~ ; consequently j-j- ~ , and-T7 3 = ^ Accordingly the proposed has neither a maximum nor a minimum value. (4) The fraction is a maximum or a minimum according as is a minimum or a maximum, as is evident from the principle that, when wis a maximum, is a minimum. But x + is a maximum or a minimum when = 1, or x= 1 . x- Again, it is easily seen that the upper sign corresponds to a mini mum and the lower to a maximum. We accordingly conclude that v> is the maximum value of , and - i its minimum value. 1+a; 3 (5) The expression u = x* has its critical value when x = . c 65. Again, to find the maximum or minimum values of u, if u=f(z), where s = <|>(:r). Here g =/W(aO, and consequently the solutions of the problem are (1) those given by <p (x) = 0, i. c. , the maximum and minimum of z ; (2) those given In many cases the values of z are restricted by the conditions of the problem to lie between given limits ; accordingly in such cases no root of f (z) = can furnish a real solution unless it lies between the given limiting values. This result will be illustrated in the following examples. 1. To find the maximum and minimum perpendicular from the focus on the tangent to an ellipse, the perpendicular p Itolnrj expressed in terms of the radius vector r. The expression for the perpendicular p, in terms of the radius vector, is Accordingly -7- = gives r= ; but these values are inadmissible, since r is restricted to lie between the values (1 +e) and (1 - c). Consequently the only maximum and minimum values of p are those which correspond to the maximum and minimum values of r; i.e., a(l + e) and (1 -e). 2. To find in an ellipse the conjugate diameters whose sum is a maximum or a minimum. If r and r be two conjugate diameters, we have r 2 + r 2 = a 2 + b 2 , The solutions accordingly are given, (1) by the maximum and minimum values of r, and (2) by the equation 1- = 0. The latter gives the equiconjugato diameters, the former the axes of the ellipse. It is easily seen that the former solution gives a maximum, the latter a minimum , -as is also readily shown other wise. 3. To find the position of a planet when brightest, its orbit and that of the earth being supposed circular, and to lie in the same plane. Let S, E, P (fig. 4) be the positions of the centres of the sun, earth, and planet respectively. Let ACBD represent the section of the planet made by the plane SEP. Draw AB perpendicular to SP, and CD perpendicular to PE. Then ADB represents the illuminated half of the planet, and CBD the half visible from the earth. Accordingly the portion of the illuminated surface turned towards the earth is contained between two planes drawn respectively through AB and CD perpendicular to SPE. This surface is projected into a crescent, the breadth of which is proportional to the versiue of BPD, or to 1 + cos EPS. Again, the brightness, depending on its distance from the earth and its position respecting the sun conjointly, will vary as 1 + cos EPS PE 2 cos EPS-- je=PE then i 2 , l + cosEPS_ " PE 2 26; Hence, neglecting a constant multiplier, we have 1 Accordingly, the solutions of the problem correspond to (1) The maximum and minimum valuesof x, i.e.,a + b and a- I; du (2) The roots of the equation ;r:, = , or of
