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Rolle’s theorem , in analysis (https://www.britannica.com/science/analysis-mathematics) , special case of the mean-value theorem (https://www.britannica.com/science/mean-value-theorem) of differential calculus (https://www.britannica.com/science/calculus-mathematics) . Rolle’s theorem states that if a function f is continuous (https://www.britannica.com/science/continuity) on the closed interval [ a , b ] and differentiable on the open interval ( a , b ) such that f ( a ) = f ( b ), then f ′( x ) = 0 for some x with a ≤ x ≤ b . In other words, if a continuous curve passes through the same y -value (such as the x -axis) twice and has a unique tangent line ( derivative (https://www.britannica.com/science/derivative-mathematics) ) at every point of the interval, then somewhere between the endpoints it has a tangent parallel to the x -axis. The theorem was proved in 1691 by the French mathematician Michel Rolle, though it was stated without a modern formal proof in the 12th century by the Indian mathematician Bhaskara II (https://www.britannica.com/biography/Bhaskara-II) . Other than being useful in proving the mean-value theorem, Rolle’s theorem is seldom used, since it establishes only the existence of a solution and not its value.

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