# Stationary point

__: Zero of the derivative of a function__

**Short description**In mathematics, particularly in calculus, a **stationary point** of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero.^{[1]}^{[2]}^{[3]} Informally, it is a point where the function "stops" increasing or decreasing (hence the name).

For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient is zero).

Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where the tangent is horizontal (i.e., parallel to the *x*-axis). For a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the *xy* plane.

## Turning points

A **turning point** is a point at which the derivative changes sign.^{[2]} A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). If the function is differentiable, then a turning point is a stationary point; however not all stationary points are turning points. If the function is twice differentiable, the stationary points that are not turning points are horizontal inflection points. For example, the function [math]\displaystyle{ x \mapsto x^3 }[/math] has a stationary point at *x* = 0, which is also an inflection point, but is not a turning point.^{[3]}

## Classification

Isolated stationary points of a [math]\displaystyle{ C^1 }[/math] real valued function [math]\displaystyle{ f\colon \mathbb{R} \to \mathbb{R} }[/math] are classified into four kinds, by the first derivative test:

- a
**local minimum**(**minimal turning point**or**relative minimum**) is one where the derivative of the function changes from negative to positive; - a
**local maximum**(**maximal turning point**or**relative maximum**) is one where the derivative of the function changes from positive to negative;

- a
**rising point of inflection**(or**inflexion**) is one where the derivative of the function is positive on both sides of the stationary point; such a point marks a change in concavity; - a
**falling point of inflection**(or**inflexion**) is one where the derivative of the function is negative on both sides of the stationary point; such a point marks a change in concavity.

The first two options are collectively known as "local extrema". Similarly a point that is either a global (or absolute) maximum or a global (or absolute) minimum is called a global (or absolute) extremum. The last two options—stationary points that are *not* local extremum—are known as saddle points.

By Fermat's theorem, global extrema must occur (for a [math]\displaystyle{ C^1 }[/math] function) on the boundary or at stationary points.

## Curve sketching

Determining the position and nature of stationary points aids in curve sketching of differentiable functions. Solving the equation *f'*(*x*) = 0 returns the *x*-coordinates of all stationary points; the *y*-coordinates are trivially the function values at those *x*-coordinates.
The specific nature of a stationary point at *x* can in some cases be determined by examining the second derivative *f''*(*x*):

- If
*f''*(*x*) < 0, the stationary point at*x*is concave down; a maximal extremum. - If
*f''*(*x*) > 0, the stationary point at*x*is concave up; a minimal extremum. - If
*f''*(*x*) = 0, the nature of the stationary point must be determined by way of other means, often by noting a sign change around that point.

A more straightforward way of determining the nature of a stationary point is by examining the function values between the stationary points (if the function is defined and continuous between them).

A simple example of a point of inflection is the function *f*(*x*) = *x*^{3}. There is a clear change of concavity about the point *x* = 0, and we can prove this by means of calculus. The second derivative of *f* is the everywhere-continuous 6*x*, and at *x* = 0, *f*′′ = 0, and the sign changes about this point. So *x* = 0 is a point of inflection.

More generally, the stationary points of a real valued function [math]\displaystyle{ f\colon \mathbb{R}^{n} \to \mathbb{R} }[/math] are those
points **x**_{0} where the derivative in every direction equals zero, or equivalently, the gradient is zero.

### Example

For the function *f*(*x*) = *x*^{4} we have *f'*(0) = 0 and *f''*(0) = 0. Even though *f''*(0) = 0, this point is not a point of inflection. The reason is that the sign of *f'*(*x*) changes from negative to positive.

For the function *f*(*x*) = sin(*x*) we have *f'*(0) ≠ 0 and *f''*(0) = 0. But this is not a stationary point, rather it is a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign of *f'*(*x*) does not change; it stays positive.

For the function *f*(*x*) = *x*^{3} we have *f'*(0) = 0 and *f''*(0) = 0. This is both a stationary point and a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign of *f'*(*x*) does not change; it stays positive.

## See also

- Optimization (mathematics)
- Fermat's theorem
- Derivative test
- Fixed point (mathematics)
- Saddle point

## References

- ↑ Chiang, Alpha C. (1984).
*Fundamental Methods of Mathematical Economics*(3rd ed.). New York: McGraw-Hill. p. 236. ISBN 0-07-010813-7. https://archive.org/details/fundamentalmetho0000chia_h4v2. - ↑
^{2.0}^{2.1}Saddler, David; Shea, Julia; Ward, Derek (2011), "12 B Stationary Points and Turning Points",*Cambridge 2 Unit Mathematics Year 11*, Cambridge University Press, p. 318, ISBN 9781107679573, https://books.google.com/books?id=wDKLXdzQL5AC&pg=PA318 - ↑
^{3.0}^{3.1}"Turning points and stationary points".*TCS FREE high school mathematics 'How-to Library'*. http://www.teacherschoice.com.au/Maths_Library/Calculus/stationary_points.htm.

## External links

- Inflection Points of Fourth Degree Polynomials — a surprising appearance of the golden ratio at cut-the-knot

Original source: https://en.wikipedia.org/wiki/Stationary point.
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