Stationary Point

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. 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 -axis). For a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the plane.

Turning points

A turning point is a point at which the derivative changes sign. 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 , which is also an inflection point, but is not a turning point.

Classification

Isolated stationary points of a $C^1$ real valued function $f\colon \mathbb \to \mathbb$ 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 $C^1$ 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 $f\colon \mathbb^ \to \mathbb$ are those
points x₀ 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

External links

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

{{Calculus topics

Differential calculus