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

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