Fermat's theorem on stationary points

Builds on Local maximum Local minimum Derivative

Statement

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$ , and let $a \in I$ . Assume that $f$ has a local maximum or a local minimum at $a$ , and that $f$ is differentiable at $a$ . Then $f'(a) = 0$ .

Remarks

At a local maximum, the difference quotient $(f(a + h) - f(a))/h$ is $\le 0$ for small $h > 0$ and $\ge 0$ for small $h < 0$ . So the limit is both $\le 0$ and $\ge 0$ . The point must be interior. On $[0, 1]$ , the function $x$ has a maximum at $1$ , where its derivative is $1$ . Mathlib states it as IsLocalExtr.deriv_eq_zero.

Reading path

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Limit point of a set definition
Local maximum definition
Limit of a function definition
Local minimum definition
Derivative definition
Fermat's theorem on stationary points statement

Where this leads

116 pages build on this one.

Tag	`0PCVBRV9VJ`
URI	`m2://0PCVBRV9VJ`
Aliases	`interior-extremum-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	26A24
Status	This copy of the site has no build data. Run `m2 status 0PCVBRV9VJ`.
Uses	Local maximum, Local minimum, Derivative
Same as	Wikidata Q727102 Wikipedia Interior extremum theorem

Fermat's theorem on stationary points

Statement

Remarks

Reading path

Where this leads

Further reading

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