. Differential and integral calculus, an introductory course for colleges and engineering schools. , (, it follows from our definition that f(a) is a mmTmum * ^^e conditions are thus shown to be sufficient. N. B. In the foregoing discussion it is tacitly assumed that/(a) ^ 366 CALCULUS §237 Problem. Assuming that f(x) and its first n + I derivatives are con-tinuous in the vicinity of x = a, employ Taylors theorem to prove thatif f(fl), f (a),f(a), . . /<s)(a) are all 0 (s («) iS|;j,then/(a)isa|-™!-What conclusion can be drawn if s is an even number? Because in deriving Taylorsformula (B
. Differential and integral calculus, an introductory course for colleges and engineering schools. , (, it follows from our definition that f(a) is a mmTmum * ^^e conditions are thus shown to be sufficient. N. B. In the foregoing discussion it is tacitly assumed that/(a) ^ 366 CALCULUS §237 Problem. Assuming that f(x) and its first n + I derivatives are con-tinuous in the vicinity of x = a, employ Taylors theorem to prove thatif f(fl), f (a),f(a), . . /<s)(a) are all 0 (s («) iS|;j,then/(a)isa|-™!-What conclusion can be drawn if s is an even number? Because in deriving Taylorsformula (B) we assumed all de-rivatives to be continuous (andtherefore finite), we cannot by this method determine the conditions for maxima and minima of the forms shown in the figure. 237. Theorems of Taylor and Maclaurin: Infinite Forms. In Taylors formula (B), Art. 229, we shall for brevity denote all theterms before R in the second member by Pn, so that the formulamay be written (B) f(a + h)=Pn+R. Pn is a polynomial of nth degree in h. Suppose we are able todetermine the numerical values of f(x),f (x), . . /(n) (x) for x = , for any assigned value of h, Pn can be determined, andformula (B) would then enable us
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