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SECTION 1.-On maxima and minima of explicit functions of one variable.

145.] Let y= F(x) be the function of which the maxima and minima are to be investigated.

From the definition it is plain that if, as x increases up to a certain value xo, F(x) increases; and afterwards as a increases, F(x) decreases; then r(x) has attained a maximum value at X = Co. And if, as ≈ increases up to a certain value ŋ, F(x) decreases, and afterwards increases as x increases, then F(x) is a minimum value.

Now Theorem I, Chapter IV, is immediately applicable to the determination of these conditions: if x and F(x) are simultaneously increasing, r'(x) is positive; if (a) decreases as x increases, F(x) is negative.

+

If therefore at any point x=xo, F'(x) changes its sign from to, we have a maximum value; and if r'(x) changes its sign from to +, we have a minimum value; and as changes of sign can take place only when the quantity passes through O or∞, we have the following rule to determine Maxima and Minima:

Find every value of x which renders r'(x) equal to O and to ∞; if such a value makes F'(x) change its sign, the corresponding value of F(x) is a maximum or minimum; being a maximum if (a) changes sign from + to, and a minimum if the change is from to +; but if there is no change of sign, there is no such singular value.

Ex. 1.

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and as F(x) is negative when x is less than a, and positive when a is greater than a, F(x) changes sign from — to + as x passes through a; and therefore F(x) has a minimum value, viz. — a2.

Ex. 2.

y = F(x) = sin x;
F′(x) = COS X = 0, if x =

π

;

2

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and as cos is positive when x is less than

2'

and negative

when a is greater than, '(x) changes sign from + to − ;

and accordingly F(x) has a maximum value, viz. 1.

Also since F(x) = cos x = 0, when x =

3п

2

from to as a passes through this value,

minimum value, viz. -1. Similarly also x =

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a change of sign from + to -, and therefore gives to sin x a maximum, viz. 1; and thus may other values be determined.

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therefore 0 is a minimum value of x (a-x)2, viz. when x = a;

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a

It is convenient to have a distinctive name for that value of a variable which makes a function of it to vanish, and therefore I propose to call it the critical value; thus a and are critical values of x in F(x) in the last example; 0 and a are the critical values of x in x (a-x)2. Of an algebraical expression, the roots are the critical values. It is plain that critical values do not necessarily cause a function to change sign, although a function cannot change sign except at a critical value at least such is the case so far as we know. Similarly the factor which vanishes is called the critical factor.

146.] When, as in Ex. 3 above, F'(x) is an algebraical function, and has many factors which, when equated to 0, cause it to vanish, it is easy to perceive their forms so that F(x) may change its sign. Corresponding to every factor of uneven dimensions, that is, of the form (x-x)2m+1, as a passes through o, there is a change of sign; but to factors of even dimensions, viz. of the form (x-x)2m, there is no change of sign as passes through ro, and therefore there is no maximum or minimum value.

To determine the change of sign corresponding to a factor of uneven dimensions, the best method is first to determine the signs of all the factors, short of the critical factor, corresponding to the critical value, and then to investigate the change of

sign of the critical factor; the following example will explain

the process.

Suppose

F(x) = x3 (x-1)2 (x-2)3 (x-4)1;

which is equal to 0, if

x = 0, and gives a change of sign from+to-, .'. a maximum;

x = 1, and gives no change of sign,

x=

=

.. no max. or minimum ;

to +, .. a minimum;

.. no max. or minimum;

x = 2, and gives a change of sign from x=4, and gives no change of sign, that is, if x 0, the critical factor is 3; but when x=0, the other factors severally are +1, -8, +16, the product of which is; and as a3, when x = 0, changes sign from to +, it follows that F(x), when x = 0, changes sign from + to, and accordingly F(x) has a corresponding maximum value; by a similar method the changes of sign due to the other critical factors are determined.

147.] Geometrical illustrations of the several conditions of maxima and minima are given in figs. 12, 13, 14, 15.

Suppose y = F(x) to represent a curve such as those drawn in the figures.

Let o M。。, and let MoPo yo the corresponding ordinate. Then in fig. 12, as x increases up to x, y = F(x) increases, and therefore F(x) is positive; but so soon as a passes the value xo, y begins to decrease, and F'(x) is negative, and the ordinate y or F(x) has manifestly attained a maximum value at x。.

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In fig. 13, the reverse is the case; as x increases up to a, y = F(x) decreases, but so soon as x is greater than x。, F(x) increases, and thus the sign of F(x) changes from to at + xo, and the corresponding value of F(x) is a minimum. Fig. 14 illustrates the case of F(x) being positive up to o and although F(x) = 0, yet it does not change its sign, but continues positive afterwards, and therefore we have no maximum value.

In the curve drawn in fig. 15, F'(x) is negative throughout; at Po it is equal to 0, but as it does not change its sign, there is no minimum value.

148.] Examples of maxima and minima.

Ex. 1. To determine the maxima and minima values of y, having given

y= F(x) = x1-8ax3 +22 a2x2-24 a3x+12 a1.

У

dy

dx

= F(x) = 4x3-24ax2+44a2x-24a3,

=

4(x-a) (x-2a) (x-3α) = 0;

if xa, and changes sign from to +,

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... a minimum;

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whence we have, if x = a, F(x) = 3a, a minimum ;

x = 2a, F(x) = 4a, a maximum;

x=3α, F(x) = 3a, a minimum.

Ex. 2. To determine the maxima and minima values of

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F′(x) = (x −1)3 (x + 2)2 (7x+5) = 0; if

x = 1, and changes sign from - to +, ... a minimum;

x=

x=

-2, but does not change sign, .. no max. or minimum;

5

and changes sign from + to -, 7

hence, when x = 1,

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... a maximum;

F(x) = 0, a minimum value;

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Ex. 3. To determine the maximum and minimum values of F(x), having given

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= 0, if x =

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-2, and does not change sign, .. no max. or min. ;

to +,.. a minimum ;

= 0, if x = 13, and changes sign from - to +,

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Ex. 4. To determine the minimum value of x*.

F(x) = xx;

F′(x) = x*(1+ log x) = 0;

if log x=-1, that is, if x = e-1; and as F'(x) changes sign from

- to +, the corresponding value of x, viz. (e), is a minimum.

Ex. 5. It is required to find the value of x, when sin x + cos x

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-

Hence the maximum value of the function is √2, and the minimum is √2; which values recur whenever a is increased by π.

149.] The change of sign of r'(x) may often be conveniently determined from the following considerations.

Let us suppose that '(x) does not vanish or become infinite

when r'(x) = 0; then since F′(x) =

d2y
dx2

=

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it is manifest that if x increases, r'(x) is positive or negative according as r'(x) simultaneously increases or decreases; but if, as a increases, F'(x) changes sign from to +, it is increasing, and if it changes sign from + to, it is decreasing; hence for a minimum value "(x) is positive, and for a maximum value r′′(x) is negative. Accordingly r(x) is a maximum or minimum value, according as the value of x, determined from the equation r'(x) = 0, renders F'(x) negative or positive.

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and F(x) = 0, if a +1, and if x = -1;

=

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