This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1922 Excerpt: ...may be greater than a maximum value of the same function. Art. 66. Conditions For Maxima And Minima By Taylor's Theorem. Let f(x) have a maximum or minimum value when x= a. Then if h be a very small increment of x, by Art. 64, /(-) f(a + h), and /(a)/(a--/t), for a maximum, also f(a)f(a + h), and f(a) f(a--h), for a ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1922 Excerpt: ...may be greater than a maximum value of the same function. Art. 66. Conditions For Maxima And Minima By Taylor's Theorem. Let f(x) have a maximum or minimum value when x= a. Then if h be a very small increment of x, by Art. 64, /(-) f(a + h), and /(a)/(a--/t), for a maximum, also f(a)f(a + h), and f(a) f(a--h), for a minimum. Therefore f(a + h)--f(a) and f(a--It)--/(a) are each negative for a maximum, or are each positive for a minimum. Now by Taylor's Theorem, f(a + h)-/(-) = fa)h +/"(a) + /"'(-)+ (1) /(a-h)-/(a) =-/'(a) h +/"(-)-/'"(-) +--(2) For a maximum: The first members of (1) and (2) must be negative, therefore the second members must be negative. Now if h be taken sufficiently small, the first term in each second member can be made numerically greater than the sum of all the terms following it; hence, the sign of each second member will be the same as that of its first term. But the first terms have different apparent signs, so the second members cannot both he negative unless the first term disappears, hence /'(a) = 0. Now the first of the remaining terms of the second members contain h2, and these terms determine the signs of the members. In order that these terms may be negative, /"(a) must be negative, or f"(a)0. Therefore, if f(a) is a maximum, f'(a) = 0 and /"(a) 0. Similarly, it may be shown that if /(a) is a minimum, fa) = 0 and f"(a) 0. However, if /"(a) = 0, then the sign of the second members of (1) and (2) will depend on the terms containing f"'(a), and since the terms containing/'"(a) have opposite signs, there can be neither a maximum nor a minimum unless /'"(a) also vanishes; and if f "'( ) = 0, then f(a) is a maximum when f"(a) is ...
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