The mean value theorem states that if a function f is continuous over the closed interval, and differentiable over the open interval (a, b), then there exists a point c in the interval (a, b) such that f'(c) is the average rate of change of the function over and it is parallel to the secant line over. Note: The result may not hold if the function is not differentiable, even at a single point in the open interval.įAQs on Mean Value Theorem What Does the Mean Value Theorem State? Thus applying the Rolles theorem, there is some x = c in (a, b) such that h'(c) = 0. H(a) = h(b) = 0 and h(x) is continuous on and differentiable on (a, b). We know that the equation of the secant line is y - y 1 = m (x - x 1). Proof: Let g(x) be the secant line to f(x) passing through (a, f(a)) and (b, f(b)). Statement: The mean value theorem states that if a function f is continuous over the closed interval, and differentiable over the open interval (a, b), then there exists at least one point c in the interval (a, b) such that f '(c) is the average rate of change of the function over and it is parallel to the secant line over.
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