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Question 7.T.13: (Cauchy’s Mean Value Theorem) If the functions f and g are c...

(Cauchy’s Mean Value Theorem)

If the functions f and g are continuous on [a, b] and differentiable on (a, b), then there is a point c ∈ (a, b) such that

[f(b) − f(a)]g^{′}(c) = [g(b) − g(a)]f^{′}(c).

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Let h be the function defined on [a, b] by

h(x) = [f(b) − f(a)]g(x) − [g(b) − g(a)]f(x),

which is clearly continuous on [a, b] and differentiable on (a, b). Since h(a) = h(b), there is a c ∈ (a, b) where

h^{′}(c) = [f(b) − f(a)]g^{′}(c) − [g(b) − g(a)]f^{′}(c) = 0

by Rolle’s theorem.

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