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 $f$.
Function of Two Variables
Let $f: \R^2 \to \R$ be a real-valued function of $2$ variables.
Let $P \in \tuple {x_0, y_0}$ be a point in $\R^2$.
$P$ is a stationary point if and only if both:
\(\ds \valueat {\dfrac {\partial f} {\partial x} } {x \mathop = x_0}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \valueat {\dfrac {\partial f} {\partial y} } {y \mathop = y_0}\) | \(=\) | \(\ds 0\) |
Also known as
A stationary point is also known (mainly in the USA) as a critical point.
Notes
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It follows from Derivative at Maximum or Minimum‎ 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.
Also see
- Results about stationary points can be found here.
Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.4$ Limits, Maxima and Minima: $3.4.2 \ (1)$
- 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): stationary point: 1a.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): stationary point (critical point): 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): stationary point (critical point): 1.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): stationary point (in one variable)