CHAPTER VIII. ON THE THEORY OF MAXIMA AND MINIMA. WHEN the variable upon which any proposed function depends passes succes. sively 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 (xh) will in the one case be both less than the maximum, and in the other both greater than the minimum. 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, (yiy) 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, d3y dx3 And in order that y may be a maximum or minimum, we must have and generally y cannot be a maximum or minimum unless the first differentia co-efficient, which does not vanish for a particular value of x be of an even order. Upon inspecting the series (1) and (2) it will be seen that, When yyare both negative, then since in this case y is greater than y', and y1, y must be a maximum; and since the whole d'y expansions are in this case negative, will have a negative sign. The re 1) equal roots each = then a, verse takes place when y is a minimum, and in this case dzy dx dy d3y has (m3) of them, and so on; till we come to da has (m2) of these roots, 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 as m is even or odd. - according 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 day or a minimum according as 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, it appears thia: y is not susceptible of a maximum or minimum value. Hence it appears that y is a maximum when x = a. y = b + (x − a)3 =0 gives in this case x = ± 1 and .: y = ± 1 and as the maximum value of y. as the minimum value of y. - 1 gives - a2 = 0 y2-2mxy + x2. dy my -x = ----- тх dx y =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 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 y = xTM (a — x)" a maximum. dy We have = xm−1 (α — x)a—1 {ma − x (m + n)} dx √3 = x2 (a — x)"−2 {(m + n − 1) (m + n) x2 — . . . . } dx2 -2 maximum which is mTM 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. |