Stationary point

Not to be confused with a fixed point where x = f(x).

This article is about stationary points or critical points of a real-valued function of one real variable. For the general notion, see Critical point (mathematics).

The stationary points are the red circles. In this graph, they are all relative maxima or relative minima. The blue squares are inflection points.

In mathematics, particularly in calculus, a stationary point or critical point of a differentiable function of one variable is a point of the domain of the function where the derivative is zero (equivalently, the slope of the graph at that point is zero). It is a point where the function "stops" increasing or decreasing (hence the name).

For a differentiable function of several variables, a stationary (critical) point is an input (one value for each variable) where all the partial derivatives are zero (equivalently, the gradient is zero).

The 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 (more specifically, parallel to the $x$ -axis). For function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the $xy$ plane.

Stationary points, critical points and turning points

The term stationary point of a function may be confused with critical point for a given projection of the graph of the function. "Critical point" is more general: a stationary point of a function corresponds to a critical point of its graph for the projection parallel to the x-axis. On the other hand, the critical points of the graph for the projection parallel to the y axis are the points where the derivative is not defined (more exactly tends to the infinity). It follows that some authors call "critical point" the critical points for any of these projections.

A turning point is a point at which the derivative changes sign.^[1] 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 $x \mapsto x^3$ has a stationary point at x=0, which is also an inflection point, but is not a turning point.^[2]

Classification

A graph in which local extrema and global extrema have been labeled.

See also: maxima and minima

Isolated stationary points of a $C^1$ real valued function $f\colon \mathbb{R} \to \mathbb{R}$ 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;

Saddle points (stationary points that are neither local maxima nor minima: they are inflection points. The left is a "rising point of inflection" (derivative is positive on both sides of the red point); the right is a "falling point of inflection" (derivative is negative on both sides of the red point).

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 $C^1$ function) on the boundary or at stationary points.

Curve sketching

The roots, turning points, stationary points, inflection point and concavity of a cubic polynomial x³ - 3x² - 144x + 432 (black line) and its first and second derivatives (red and blue).

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) = x³. 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 6x, 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 $f\colon \mathbb{R}^{n} \to \mathbb{R}$ are those points x₀ where the derivative in every direction equals zero, or equivalently, the gradient is zero.

Example

For the function f(x) = x⁴ 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) = x³ 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.

References

↑ 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
↑ "Turning points and stationary points". TCS FREE high school mathematics 'How-to Library',. Retrieved 30 October 2011.

External links

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

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