and negative in the latter case; and this must be true ultimately

when dx is diminished without limit. Hence must be

du
dx

positive in the former and negative in the latter case.

du

Su

Conversely, since is the limiting value of it is evident

dx

be positive or negative when dx is sufficiently small, and that consequently u must be increasing or decreasing with the

du dx

increase of x accordingly as is positive or negative.

In other words, that u may be increasing or decreasing with du an increase of x, it is sufficient and necessary that be positive in the former and negative in the latter case.

Rule for finding Maxima and Minima.

dx

70. By the definition of a maximum or minimum given above, and by virtue of the Lemma of the preceding article, we see that, for a maximum value of y, it is sufficient and

dy dx

necessary that change sign from to as x passes from

x-h to x + h, being positive for the former range of values of x and negative for the latter. From these sufficient and

necessary conditions it follows that, when x = x

dy

dx

must

become either zero or infinity, since a function of x can change sign only in passing through one or other of these values. Similarly, that x = x, may correspond to a minimum value of y,

it appears that must pass from to as x passes from

dy
dx

h

to x + h and become, as in the case of a maximum, either zero or infinity when x = x.

We may now enunciate a general rule for finding the maximum and minimum values of y.

RULE. Obtain all the values of x which satisfy either of the

two equations

f'(x) = 0,

ƒ'(x) = ∞ :

+

if any one of these values of x be such that, as 2 increases through it, ƒ'(x) changes sign from to a maximum value of y; if it be through it, f'(x) changes sign from

to -, it will correspond such that, as x increases

to +, it will correspond

to a minimum value of y; and if it be such that, as x increases through it, f'(x) does not change sign, it will correspond neither to a maximum nor to a minimum value of y.

Ex. 1. To find whether, n being a positive integer,

dy dx

will be negative in the former

dy case and positive in the latter; and that, if n be odd, will dx

have the same sign in both cases. Hence, if n be even, x = ά gives zero as a minimum value of y, and, if n be odd, y has neither a maximum nor a minimum value.

71. Suppose (x) to be any function of x which for all possible values of x has the same sign as ƒ'(x). Then it is evident that in the rule which we have enunciated for finding the maximum and minimum values of f(x), we may replace f'(x) by p(x). This change will frequently abbreviate the processes of investigation. Thus if, for instance,

ƒ'(x) = ¥(x). p(x),

where (x) is a function of x essentially positive, we may reject (x) and take p(x) in place of ƒ'(x).

Ex. 1. To find the maximum or minimum values of y, when

now (x) cannot change sign: put therefore

it will be observed that we escape the trouble of examining the consequences of putting f'(x) = ∞.

Alternation of Maxima and Minima.

72. Supposing y to be a function f(x) of x, which has several maxima and minima, then, as x keeps continuously

increasing, the maximum and minimum values of y will occur alternately. This will easily be seen when we consider that whenever the sign of f'(x) changes from + to

-, y is a maximum, and, whenever it changes from to +, a minimum; and that a change from to can be succeeded only by a change from to +, and conversely.

Modified method of finding Maxima and Minima.

73. Suppose that f(x) has a maximum value when x = x。, and that none of the derived functions

و

ƒ'(x), ƒ"(x), ƒ''''(x),

Then, since f'(x) decreases

==

become infinite when x = = x。・ from, through 0, to, as a passes from x-h, through x, to xh, it appears, by the Lemma of Art. (69), that ƒ"(x) must have the sign – for this range of values of x. If ƒ"(x) = −0, when x = x, the symbol - 0 being used to denote zero regarded as a limiting state of negative magnitude, then, when x = x, it is evident that f(x) has a maximum value: from this it follows that, since f(x) now occupies the place of f(x), ƒ''(x) will change sign from +, through 0, to, and f'(x) will have the sign, as a ranges from xh to x + h. If f" (x)· 0, when x = x, then, f'(x) now occupying the place of f(x), we see that f(x) will change sign from+, through 0, to, and ƒ''(x) will have the sign as x ranges from x h to xo + We may proceed with this reasoning from step to step until we arrive at a derived function of an even order which does not vanish when x = xo. Our final conclusion is evidently that, for a maximum value of f(x), we must have ƒ'(x) = 0, and that, of the differential coefficients of f(x), the first which, for a corresponding value of x, does not vanish, must be of an even order, and must be negative.

,

h.

By precisely the same form of reasoning, mutatis mutandis, we may see that, for a minimum value of f(x), the sufficient and necessary conditions are that

ƒ' (x) = 0,