Cauchy's Mean-Value Theorem (The Generalized Mean-Value Theorem)
設f(x)、g(x)在[a,b]連續，在(a,b)可微分，則存在一個c€(a,b)使得
f'(c)[ g(b) - g(a) ] = g'(c)[ f(b) - f(a) ]
pf：
令h(x) = f(x)[g(b) - g(a)] - g(x)[f(b) - f(a)] ， x€[a,b]
則h(x)在[a,b]連續，在(a,b)可微分，
且h'(x) = f'(x)[g(b) - g(a)] - g'(x)[f(b) - f(a)]，
又因為h(a) = f(a)[g(b) - g(a)] - g(a)[f(b) - f(a)] = f(a)g(b) - g(a)f(b)
h(b) = f(b)[g(b) - g(a)] - g(b)[f(b) - f(a)] = f(a)g(b) - g(a)f(b)
所以 h(a) = h(b)
則根據 Mean-Value Theorem ，存在一個c€(a,b) 使得 h'(c) = 0
→ f'(c)[ g(b) - g(a) ] = g'(c)[ f(b) - f(a) ]
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※ 編輯: jellyfishing 來自: 123.194.60.211 (12/19 20:05)