Stationary point

From Example Problems
(Redirected from Extrema)
Jump to: navigation, search
File:Stationary vs inflection pts.gif
Stationary points (red pluses) and inflection points (green circles). The stationary points in this graph are all relative maxima or relative minima.

In mathematics, particularly in calculus, a stationary point is a point on the graph of a function where the tangent to the graph is parallel to the x-axis (in 2D) or the plane tangent to the surface is parallel to the XY plane (in 3D). An equivalent definition is where the derivative of the function equals zero (known as a critical number).

File:Stationary and inflection pts.gif
Saddle points (coincident stationary points and inflection points). Here one is rising and one is a falling inflection point.

An inflection point is a point where the concavity changes. A point of inflection is not necessarily a stationary point. All inflection points have the property of f''(x) = 0 but the reverse is not necessarily true.

Stationary points of a real valued function f: RR are classified into four kinds:

  • a minimal extremum (minimal turning point or relative minimum) is one where the derivative of the function changes from negative to positive;
  • a maximal extremum (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

Notice: Global (or absolute) maxima and minima are sometimes called global (or absolute) maximal (resp. minimal) extrema. While they may occur at stationary points, they are not actually an example of a stationary point.

Determining the position and nature of stationary points aids in curve sketching, especially for continuous 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 a maximal extremum.
  • If f''(x) > 0, the stationary point at x is 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 provided the function values exist 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. However, this is limited again in that it works only for functions that are continuous in at least a small interval surrounding the stationary point.

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: RnR are those points x0 where the derivative in every direction equals zero, or equivalently, the gradient is zero.

Example

At x1 we have f' (x) = 0 and f''(x) = 0. Even though f''(x) = 0, this point is not a point of inflexion. The reason is that the sign of f' (x) changes from negative to positive.

At x2, we have f' (x) \neq 0 and f''(x) = 0. But, x2 is not a stationary point, rather it is a point of inflexion. This because the concavity changes from concave upwards to concave downwards and the sign of f' (x) does not change; it stays positive.

At x3 we have f' (x) = 0 and f''(x) = 0. Here, x3 is both a stationary point and a point of inflexion. This is because the concavity changes from concave upwards to concave downwards and the sign of f' (x) does not change; it stays positive.

See also

External link