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:Let f : [a , b] -> R be continuous on the interval [a , b] and differentiable on (a , b) and suppose that f(a) = f(b). Then there exists some x in (a , b) such that ::f ' (x) = 0 |
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:Let f : [a , b] -> R be continuous on the interval [a , b] and differentiable on (a , b) and suppose that f(a) = f(b). Then there exists some x in (a , b) such that f ' (x) = 0. |
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Generalization: The theorem is usually stated in the form above, but it is actually valid in a slightly more general setting: We only need to assume that f : [a , b] -> R is continuous on [a , b], that f(a) = f(b), and that for every x in (a , b) the limit limh->0 (f(x+h)-f(x))/h exists or is equal to +/- infinity. |
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