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A Discussion On Rolle's Theorem

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Please solve the following with as much explanation of each step as possible.

Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Also, suppose that f(a) = f(b) and that c is a real number in the interval such that f'(c) = 0. Find an interval for the function g over which Rolle's theorem can be applied, and find the corresponding critical number of g (k is a constant).

(a) g(x) = f(x) + k

(b) g(x) = f(x-k)

(c) g(x) = f(kx)

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Solution Summary

In this solution, the method to find the interval on which one can apply Rolle's theorem to a continuous and differentiable function is discussed. Step by step calculations are given.

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Problem: Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Also, suppose that f(a) = f(b) and that c is a real number in the interval such that f'(c) = 0. Find an interval for the function g over which Rolle's theorem can be applied, and find the corresponding critical number of g (k is a constant).

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