Exercise 7: Let a < b. Suppose that f and g are two functions that are continuous on [a, bl, and differentiable on (a, b). Further suppose that g' has no zeroes on (a, b). Prove that there exists c e (a, b) such that f (b) f(a) g(b) g(a) f'(c) g'(c) Hint: Consider F(x) = [f(b) - f(a)]g(x) - [g(b)- g(a)] f (x). Remark: This is a generalization of the mean value theorem, called the Cauchy mean value theorem Note if g(x)= x, then this is the mean value theorem.

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A: This is a consequence of the intermediate value theorem

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