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Real function/Open interval/Local extrema/Differentiable/Derivative zero/Fact/Proof

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Proof
Proof

We may assume that attains a local maximum in . This means that there exists an , such that holds for all . Let be a sequence with , tending to ("from below“). Then , and so , and therefore the difference quotient

Due to fact, this relation carries over to the limit, which is the derivative. Hence, . For another sequence with , we get

Therefore, also and thus .