# Definition:Stationary Point

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## Definition

Let $f$ be a real function which is differentiable on the open interval $\openint a b$.

Let $\exists \xi \in \openint a b: \map {f'} \xi = 0$, where $\map {f'} \xi$ is the derivative of $f$ at $\xi$.

Then $\xi$ is known as a **stationary point of $f$**.

## Also known as

A **stationary point** is also known (mainly in USA) as a **critical point**.

## Notes

It follows from Derivative at Maximum or Minimumâ€Ž that any local minimum or local maximum is a stationary point.

However, it is not the case that a stationary point is always either a local minimum or local maximum.

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 11.3$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**critical point**:**1.** - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**stationary point**:**1a.** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**stationary point (critical point)**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**stationary point (critical point)**:**1.** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**stationary point**(in one variable)