Differentiable function/R/Local extrema/Derivative/Introduction/Section

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be a function which attains in a local extremum, and is differentiable there. Then holds.

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 .

We remark that the vanishing of the derivative is only a necessary, but not a sufficient, criterion for the existence of an extremum. The easiest example for this phenomenon is the function , which is strictly increasing and its derivative is zero at the zero point. We will provide a sufficient criterion in fact, see also fact.