If Taylor's theorem fails to give the expansions of ƒ (a +- h), 4 (a + h), which will happen wherever ƒ (x), or ø (x) contains a radical which vanishes for x = a, we must obtain the expansion by some other method.

Substitute. (a + h) for x in both terms of the fraction, and developing by binomial theorem, we shall have

Dividing both the numerator and denominator by the lowest power of h Now make h=0

It is useless to take the differential co-efficients of the terms in this case, hecause they become infinite. Making x = a + h, we find for h = 0

We may here employ Taylor's theorem to determine those terms of the series for which it holds good, we shall thus obtain upon substituting c + k

for x

√h + h√ c = b + ···

vh — } h (2c)

dividing by √h and then making h=0 we find 1 for the true value of the

fraction.

Ex. 4. When x = a

When x = a gives a product ƒ (x) × ọ (x) of the form 0 × ¤,

f (x)
Q(x)

becomes

on the supposition that a = a, but by the above process we shall have

818

CHAPTER VIII.

ON THE THEORY OF MAXIMA AND MINIMA.

WHEN the variable upon which any proposed function depends passes successively through all degrees of magnitude, the different values of the function may form first an increasing and then a decreasing series, or vice versa, and may go on increasing or decreasing repeatedly, and vice versa.

That value at which an increase of the function ends, and a decrease begins, is called a maximum, and that at which a diminution ends, and an increase begins, is called a minimum.

The essential characteristic of a maximum consists in its being greater than each of the values of the function which immediately precede and follow it; and that of a minimum in being less than both these values.

Let y be any function of x in which this variable has attained a value which constitutes it either a maximum or a minimum. Then if x be increased and diminished by an indefinitely small quantity h, the developements of (x + h) and (x h) will exhibit the values of x, immediately adjacent on each side to that value which renders y a maximum or minimum. Hence it follows from our definition, that the values of y corresponding to (x + h) and (x h) will in the one case be both less than the maximum, and in the other both greater than the minimum.

Let

Now, in order that y may be a maximum or minimum, the values of y' and y which immediately precede and follow it must be both less or both greater than y.

.. When y is a maximum or minimum, (y' — y) and (y, — y) must both have the same sign.

But when his assumed infinitely small, the whole of the expansions (1) and (2) will have the same signs as their first terms.

Hence, (yi y) and (y'

- y) cannot have the same sign, unless p vanishes, ... in order that y may be a maximum or a minimum, the condition requisite is that,

d'y

dx3 = 0,

And in order that y may be a maximum or minimum, we must have nd generally y cannot be a maximum or minimum unless the first differential o-efficient, which does not vanish for a particular value of x be of an even rder.

Upon inspecting the series (1) and (2) it will be seen that,

When

y' yare both negative, then since in this case y is greater than Yy1 yy, and y1, y must be a maximum; and since the whole expansions are in this case negative, will have a negative sign. The re

d'y dx

verse takes place when y is a minimum, and in this case

1) equal roots each =

then d,

dx

d3y

has (m3) of them, and so on; till we come to

dam

has (m —2) of these roots, d which is the first differential co-efficient which does not contain the root, and in this case, the values y', y1, corresponding to (x + h), and (x — h), are

The sign of the second term in this last expansion being + or - according as m is even or odd. Hence (y' — y), and (y, — y) cannot have the same sign if m be odd, and .. in this case y is neither a maximum nor minimum. But if m be even, then (y' — y) and (y, — y) will have the same sign, and y is a maximum

or a minimum according as

dy dxTM

is positive or negative.

Ex. 1. Let y = √2mx

Required to determine the value of x which will render y a maximum or minimum.

Since we should obtain no result by equating this quantity to 0, y is not susceptible of a maximum or minimum value.

Hence it appears that y is a maximum when x = a.

= 0 gives in this case x = 1 and .: y = + 1⁄2 and

x=

1 gives

as the maximum value of y. - as the minimum value of y.

y2 -- 2mxy + x2 a2 = 0

=0 we have my = x, and eliminating x and y by the original

Ex. 6. To divide a given number a into two parts, so that the product of the mth power of the one multiplied by the nth power of the other, shall be the greatest possible.

Let x be one of the parts, and let y be the product of the two parts; then it is required to find the value of x which will render the quantity

d2y

1-2

da2

=xm2 (α — x)"−2 {(m + n − 1) (m + n) x2 — . . . .}

-..

ma

dy dx

Putting =0 we have x 0, x = a, x =

this last root gives the

m+n'

maximum which is mm n" (m + n)"+".

The two other roots correspond to the minima when m and n are equal.

A great number of interesting geometrical problems may be solved by the application of these principles. The following are a few examples.