Quod Erat Demonstrandum

2010/07/26

MVT 某推廣

Filed under: HKALE,Pure Mathematics — johnmayhk @ 11:54 上午
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查 THE COLLEGE MATHEMATICS JOURNAL VOL. 35 NO.1 JANUARY 2004 有關 Cauchy’s Mean Value Theorem Involving n Functions 一文,當中以

證明

F(x) = -3x + \frac{\pi}{2}\cos(\frac{\pi}{2}x) + \frac{e^x}{e - 1} + \frac{1}{(x + 2)\ln 2} 在區間 (0,1) 存在零點 ………. (*)

時,

F(0) , F(1) 皆大於零,故不宜用介值定理(Intermediate Value Theorem)。

F(1) \approx -0.696675773,即可用介值定理一步 KO 的。

文中的【主要定理】如下:

\alpha_1, \alpha_2, \dots , \alpha_n 為 n 個實數,且有 \alpha_1 + \alpha_2 + \dots + \alpha_n = 0

f_1(x), f_2(x), \dots , f_n(x) 為 n 個在 [a , b] 上連續並在 (a , b) 上可導之函數,

且對於 i = 1, 2, \dots , n,有 f_i(a) \ne f_i(b)

則在 (a , b) 中存在 c 值使下式成立:

\displaystyle\sum_{i = 1}^n \frac{\alpha_i f_i'(c)}{f_i(b) - f_i(a)} = 0

為應用上述定理,作者取

n = 4
a = 0
b = 1
f_1(x) = x^2
f_2(x) = \sin(\frac{\pi}{2}x)
f_3(x) = e^x
f_4(x) = \ln(x + 1)
\alpha_1 = -3, \alpha_2 = \alpha_3 = \alpha_4 = 1

從而得出 (*)。

現考慮【主要定理】在 n = 3 的情況下之證明。

g(x)
= \alpha_1(f_2(b) - f_2(a))(f_3(b) - f_3(a))(f_1(x) - f_1(a))
+ \alpha_2(f_1(b) - f_1(a))(f_3(b) - f_3(a))(f_2(x) - f_2(a))
+ \alpha_3(f_1(b) - f_1(a))(f_2(b) - f_2(a))(f_3(x) - f_3(a))

易知

g(a) = 0

g(b) = (f_1(b) - f_1(a))(f_2(b) - f_2(a))(f_3(b) - f_3(a))(\alpha_1 + \alpha_2 + \alpha_3) = 0

由羅氏定理(Rolle’s Theorem),得到結論:在 (a , b) 中存在 c 值使 g'(c) = 0

result follows.

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