# Stationary point

A **stationary point** of a differentiable function is a point in its domain at which the derivative vanishes, so the tangent to the graph is horizontal. For a function \( f(x) \) of one variable, it is a point \( x^* \) where \( f'(x^*) = 0 \); such a point is also called a critical point.<sup>[1](https://www.sciencedirect.com/topics/engineering/stationary-point)</sup><sup> • </sup><sup>[2](https://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec3.pdf)</sup> Stationary points are the starting place of optimization in calculus, because a differentiable function can attain a local maximum or minimum only at such a point, although not every stationary point is an extremum.

| Key fact | Detail |
|---|---|
| Definition | A point \( x^* \) in the domain of a continuous, differentiable function \( f \) at which \( f'(x^*) = 0 \); the tangent is horizontal<sup>[1](https://www.sciencedirect.com/topics/engineering/stationary-point)</sup> |
| Other names | Critical point; a point where \( f'(t_0) = 0 \) is a stationary point, but it need not be a local maximum or minimum<sup>[2](https://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec3.pdf)</sup><sup> • </sup><sup>[3](https://encyclopediaofmath.org/wiki/Gradient)</sup><sup> • </sup><sup>[18](https://www.mas.ncl.ac.uk/ask/numeracy-maths-statistics/core-mathematics/calculus/stationary-points.html)</sup> |
| Types (one variable) | Local maximum, local minimum, or stationary point of inflection; maxima and minima are called turning points<sup>[2](https://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec3.pdf)</sup><sup> • </sup><sup>[4](https://www.open.edu/openlearn/science-maths-technology/introduction-differentiation/content-section-6.2)</sup> |
| Second derivative test | \( f'' > 0 \) gives a minimum, \( f'' < 0 \) a maximum, \( f'' = 0 \) is inconclusive<sup>[2](https://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec3.pdf)</sup> |
| Several variables | All first-order partial derivatives vanish, \( \nabla f = 0 \); points are classified as maximum, minimum, or saddle by the Hessian<sup>[5](https://openstax.org/books/calculus-volume-3/pages/4-7-maxima-minima-problems)</sup> |
| Fermat's theorem | A local extremum of a differentiable function must occur at a stationary point; the condition is necessary but not sufficient<sup>[6](https://github.com/antoniolupetti/algebrica/blob/main/differential-calculus-theorems/fermat-theorem.md)</sup> |
| History | Pierre de Fermat found maxima and minima by locating where the tangent is parallel to the x-axis, and stated a second-derivative criterion in 1643<sup>[7](https://mathshistory.st-andrews.ac.uk/DSB/Fermat.pdf)</sup> |

## Definition and how stationary points are found

The derivative \( dy/dx \) measures the rate of change of \( y \) with respect to \( x \); when it is zero, the gradient of the graph is zero and the tangent to the curve is horizontal.<sup>[2](https://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec3.pdf)</sup> To find the stationary points of a function given by a formula, the standard procedure is to differentiate, set the derivative equal to zero, and solve the resulting equation \( dy/dx = 0 \).<sup>[8](https://www.met.reading.ac.uk/pplato2/h-flap/math4_4.html)</sup> Substituting each solution back into the function gives the full point on the graph.

For example, for \( f(x) = 2x^3 + 3x^2 - 12x + 17 \), setting \( f'(x) = 6x^2 + 6x - 12 = 0 \) gives \( x = 1 \) and \( x = -2 \), so the stationary points are \( (1, 10) \) and \( (-2, 37) \).<sup>[9](https://www.sydney.edu.au/content/dam/students/documents/mathematics-learning-centre/introduction-to-differential-calculus.pdf)</sup> A function can have many stationary points; \( \sin x \) has infinitely many maxima and minima.<sup>[2](https://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec3.pdf)</sup>

## Classification in one variable

**Three types.** Moving from left to right through a stationary point, the gradient may change from positive to negative, giving a local maximum; from negative to positive, giving a local minimum; or keep the same sign on both sides, giving a point of inflection.<sup>[2](https://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec3.pdf)</sup> The curve \( y = x^3 \) at \( x = 0 \) is the standard example of the third kind: the gradient is zero there but positive everywhere else.<sup>[2](https://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec3.pdf)</sup> Local maxima and minima are called turning points, because the graph stops rising or falling and turns the other way.<sup>[4](https://www.open.edu/openlearn/science-maths-technology/introduction-differentiation/content-section-6.2)</sup>

**The second derivative test.** If \( dy/dx = 0 \) at \( x = x_0 \), then \( d^2y/dx^2 > 0 \) at \( x_0 \) means the stationary point is a minimum, \( d^2y/dx^2 < 0 \) means it is a maximum, and \( d^2y/dx^2 = 0 \) means the test fails and further investigation is needed.<sup>[2](https://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec3.pdf)</sup> The first derivative test is an alternative: choose sample points on each side of the stationary point and compare the signs of the derivative, with positive-then-negative indicating a local maximum, negative-then-positive a local minimum, and equal signs a horizontal point of inflection.<sup>[4](https://www.open.edu/openlearn/science-maths-technology/introduction-differentiation/content-section-6.2)</sup>

A higher-order test covers cases the second derivative test misses. If \( f \) has \( m \) derivatives at \( x_0 \) with \( f^{(k)}(x_0) = 0 \) for \( k = 1, \dots, m-1 \) and \( f^{(m)}(x_0) \neq 0 \), then \( x_0 \) is a strict local maximum if \( m \) is even and \( f^{(m)}(x_0) < 0 \), and a strict local minimum if \( m \) is even and \( f^{(m)}(x_0) > 0 \).<sup>[10](https://encyclopediaofmath.org/wiki/Maximum_and_minimum_points)</sup>

## Relation to extrema

By Fermat's theorem, a differentiable function can attain a local extremum—that is, a relative maximum or minimum—inside its domain only at a stationary point, meaning a point at which the first derivative vanishes and the tangent line is horizontal.<sup>[6](https://github.com/antoniolupetti/algebrica/blob/main/differential-calculus-theorems/fermat-theorem.md)</sup> The condition is necessary but not sufficient: a zero derivative does not necessarily imply an extremum, as \( f(x) = x^3 \) shows, since \( f'(0) = 0 \) but the function is increasing everywhere.<sup>[6](https://github.com/antoniolupetti/algebrica/blob/main/differential-calculus-theorems/fermat-theorem.md)</sup><sup> • </sup><sup>[11](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima)</sup>

Stationary points are also not the only candidates for global extrema. A function can reach its greatest and least values at discontinuities, at the boundaries of its domain, or at critical points where its derivative vanishes.<sup>[12](https://www.ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/pages/unit-2-applications-of-differentiation/part-b-optimization-related-rates-and-newtons-method/session-30-optimization-problems-ii/)</sup> For a continuous function on a closed, bounded interval, the absolute maximum and minimum must occur at endpoints or at critical points.<sup>[11](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima)</sup>

## Several variables

For a function \( f(x, y) \) of two variables, a stationary point (also called a critical point) is a point where both first-order partial derivatives vanish, \( f_x = f_y = 0 \); if only one of them is zero, the point is not a critical point.<sup>[13](https://tutorial.math.lamar.edu/classes/calciii/relativeextrema.aspx)</sup> Geometrically, the tangent plane is horizontal at such a point.<sup>[14](https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/2df7f490fc5f1bd28d6e8fd391e9a964_MIT18_02SC_notes_16.pdf)</sup> As in one variable, not every stationary point is an extremum: for \( f(x, y) = x^2 - y^2 \), both partial derivatives vanish at the origin, which is a saddle point, a global maximum along one trace and a global minimum along the other.<sup>[5](https://openstax.org/books/calculus-volume-3/pages/4-7-maxima-minima-problems)</sup>

The second derivative test for two variables uses the discriminant \( D = f_{xx}(x_0, y_0) f_{yy}(x_0, y_0) - \left( f_{xy}(x_0, y_0) \right)^2 \), computed at each critical point, to classify it as a local maximum, local minimum, or saddle point, or the test may be inconclusive.<sup>[5](https://openstax.org/books/calculus-volume-3/pages/4-7-maxima-minima-problems)</sup> A critical point at which the [Hessian matrix](https://www.edgepedia.org/hessian-matrix) is singular is called degenerate; a nondegenerate critical point, one whose Hessian is nonsingular, is isolated, meaning a neighborhood exists containing no other critical point, although some isolated critical points are degenerate.<sup>[15](https://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node139.html)</sup>

## Optimization and numerical methods

Finding stationary points is the core of unconstrained optimization. Solving the equations \( \nabla f(x) = 0 \) analytically is often impractical, because the equations are usually nonlinear and there may be infinitely many stationary points, so iterative algorithms are used instead.<sup>[16](https://epubs.siam.org/doi/book/10.1137/1.9781611977622)</sup> [Newton's method](https://www.edgepedia.org/newtons-method) minimizes the quadratic approximation of the function around each iterate; when the Hessian \( \nabla^2 f(x_k) \) is positive definite, the update \( x_{k+1} = x_k - \left( \nabla^2 f(x_k) \right)^{-1} \nabla f(x_k) \) moves to the unique stationary point of that quadratic model.<sup>[16](https://epubs.siam.org/doi/book/10.1137/1.9781611977622)</sup>

## History

[Pierre de Fermat](https://www.edgepedia.org/pierre-de-fermat) investigated maxima and minima by considering when the tangent to a curve is parallel to the x-axis, essentially the method used today of calculating where the derivative of a function is zero; because of this work, Lagrange stated that he considered Fermat the inventor of the calculus.<sup>[17](https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/)</sup> In a 1643 demonstration, Fermat established what is now termed the second derivative criterion for the nature of an extreme value, \( f''(x) < 0 \) for a maximum and \( f''(x) > 0 \) for a minimum, although he did not investigate points of inflection, where \( f''(x) = 0 \).<sup>[7](https://mathshistory.st-andrews.ac.uk/DSB/Fermat.pdf)</sup>

## References

1. [Stationary Point - an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/engineering/stationary-point)
2. [Applications of the Derivative, 1C2 Section 3, University of Manchester (D. Harris)](https://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec3.pdf)
3. [Gradient, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Gradient)
4. [Introduction to differentiation: 4.2 Stationary points, OpenLearn, The Open University](https://www.open.edu/openlearn/science-maths-technology/introduction-differentiation/content-section-6.2)
5. [4.7 Maxima/Minima Problems, Calculus Volume 3, OpenStax](https://openstax.org/books/calculus-volume-3/pages/4-7-maxima-minima-problems)
6. [Fermat's Theorem, Algebrica](https://github.com/antoniolupetti/algebrica/blob/main/differential-calculus-theorems/fermat-theorem.md)
7. [Pierre de Fermat, Dictionary of Scientific Biography, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/DSB/Fermat.pdf)
8. [PPLATO FLAP MATH 4.4: Stationary points and graph sketching, University of Reading](https://www.met.reading.ac.uk/pplato2/h-flap/math4_4.html)
9. [Introduction to differential calculus, Mathematics Learning Centre, University of Sydney](https://www.sydney.edu.au/content/dam/students/documents/mathematics-learning-centre/introduction-to-differential-calculus.pdf)
10. [Maximum and minimum points, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Maximum_and_minimum_points)
11. [4.3 Maxima and Minima, Calculus Volume 1, OpenStax](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima)
12. [Session 30: Optimization Problems II, MIT OpenCourseWare](https://www.ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/pages/unit-2-applications-of-differentiation/part-b-optimization-related-rates-and-newtons-method/session-30-optimization-problems-ii/)
13. [Calculus III - Relative Minimums and Maximums, Paul's Online Math Notes](https://tutorial.math.lamar.edu/classes/calciii/relativeextrema.aspx)
14. [18.02SC Notes: Critical Points, MIT OpenCourseWare](https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/2df7f490fc5f1bd28d6e8fd391e9a964_MIT18_02SC_notes_16.pdf)
15. [7.3.2 Nonisolated stationary points, MIT Hyperbook (Patrikalakis, Maekawa, Cho)](https://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node139.html)
16. [Introduction to Nonlinear Optimization, 2nd ed., SIAM Publications Library](https://epubs.siam.org/doi/book/10.1137/1.9781611977622)
17. [The rise of calculus, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/)
18. [mas.ncl.ac.uk](https://www.mas.ncl.ac.uk/ask/numeracy-maths-statistics/core-mathematics/calculus/stationary-points.html)

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