. Differential and integral calculus, an introductory course for colleges and engineering schools. f course it is heretacitly assumed that f(x), g{x), and their first n + 1 derivativesare continuous in the vicinity of x = a. 236. Criteria for Maxima and Minima Determined by TaylorsTheorem. Lemma 1. In the polynomial bQ + bih + b2h2 + bji3 + • • • + bkh\ h can be made so small that the polynomial shall have the sign ofits first term bo. For, since the 6s are finite constants, and there is a finite numberof them, h can be made so small that bih + b2h2 + bsh3 + • • • + bkhk = h(bi + b2h + • • - +
. Differential and integral calculus, an introductory course for colleges and engineering schools. f course it is heretacitly assumed that f(x), g{x), and their first n + 1 derivativesare continuous in the vicinity of x = a. 236. Criteria for Maxima and Minima Determined by TaylorsTheorem. Lemma 1. In the polynomial bQ + bih + b2h2 + bji3 + • • • + bkh\ h can be made so small that the polynomial shall have the sign ofits first term bo. For, since the 6s are finite constants, and there is a finite numberof them, h can be made so small that bih + b2h2 + bsh3 + • • • + bkhk = h(bi + b2h + • • - + bkhk~l) shall be less than any quantity that can be named, and hence itcan be made less than &o. And then the polynomial will have thesign of 60-Lemma 2. In the polynomial bohm + bihm+l + b2hm+2+ • • • + bkhm+k, h can be made so small that the polynomial shall have the sign of itsfirst term, bohm. For, the polynomial may be written in the form hm(bo + bih + b2h2 + ■ ■ • +bkhk), and by lemma 1, the polynomial within the parentheses can be made 364 CALCULUS §236. to have the sign of 60, by making h small enough, and then thegiven polynomial will have the sign of its first term, bohm. A study of the accompanying figures will make it plain that ourdefinitions of maximum and minimum, Art. 59, are equivalentto the following Definitions. If f(a + h) — f(a) and f(a — h) — f(a) are both , j for all values of h within a sufficiently small interval, PQ,that contains x = a, then /(a) maximumminimum We have now to prove thefollowing Theorem. Assuming that f(x),f(x), f(x), and f(x) are con-tinuous throughout an intervalthat contains x = a, the necessaryand sufficient conditions that f(a) (maximumI minimum is P a q Y .1 ^^Wl 0 f(«> X i 3 a />>= °> andW is j; We develop f(a + h) to four terms by Taylors formula (B), orby (30 of Art. 227, and then transpose f(a) to the first member ofthe equat
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