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Rolle's theorem states that if a polynomial function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), and if f(a) = f(b), then there exists at least one point c in (a,b) such that f'(c) = 0.

In simpler terms, Rolle's theorem tells us that if a polynomial function has the same value at its endpoints, then there must be at least one point in between where the derivative is equal to zero.

To prove Rolle's theorem, we start by assuming that f(x) is continuous on [a,b] and differentiable on (a,b), and that f(a) = f(b). We then consider two cases:

Case 1: f(x) is constant on [a,b]. In this case, f'(x) = 0 for all x in (a,b), so we can choose any point c in (a,b) and f'(c) = 0.

Case 2: f(x) is not constant on [a,b]. In this case, we know that f(x) must have a maximum or minimum value on (a,b) by the extreme value theorem. Let c be a point in (a,b) where f(c) is a maximum or minimum value. Since f(x) is differentiable on (a,b), we know that f'(c) = 0.

Therefore, in either case, we have shown that there exists at least one point c in (a,b) such that f'(c) = 0, which proves Rolle's theorem.

Rolle's theorem is useful in many applications, such as finding the roots of a polynomial function or proving the existence of critical points in optimization problems.

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