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Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and...

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Advance Math

Rolle's Theorem, "Let f be a continuous function on [a,b] thatis differentiable on (a,b) and such that f(a)=f(b). Then thereexists at least one point c on (a,b) such that f'(c)=0."

Rolle's Theorem requires three conditions be satisified.

(a) What are these three conditions?

(b) Find three functions that satisfy exactly two of these threeconditions, but for which the conclusion of Rolle's theorem doesnot follow, i.e., there is no point c in (a,b) such that f'(c)=0.Each function should satisfy a different pair of conditions thanthe other two functions. For each function you should give adefinition, a graph, and a short justification of its failing tomeet the conclusion of Rolle's Theorem.

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