Understanding Rolle’s Theorem: Exploring the Fundamental Concept in Calculus

Rolle’s Theorem

If f(x) is continuous on the closed interval [a, b], differentiable on (a, b), and satisfies f(a) = f(b), then for some c in the interval (a, b), we have f'(c) = 0

Rolle’s Theorem is a fundamental theorem of calculus that deals with differentiable functions and their derivatives. It states that for a differentiable function f(x) defined on a closed interval [a,b], if the value of f(a) is equal to the value of f(b), then there must be at least one point c, where a < c < b, where the derivative of the function f(x) is equal to zero. In simpler terms, Rolle's Theorem states that if a continuous curve starts and ends at the same height, then there must be at least one point on the curve where its tangent is horizontal. The theorem was first proven by the French mathematician Michel Rolle in 1691, hence the name Rolle's Theorem. It is an important theorem in calculus as many other theorems rely on it for their proof.

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