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Stationary point

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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 or, equivalently, where the derivative of the function equals zero.

Stationary points are classified into four kinds:

  • a minimal extremum (or minimal turning point) is one where the derivative of the function changes from negative to positive;
  • a maximal extremum (or maximal turning point) 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;
  • a falling point of inflection (or inflexion) is one where the derivative of the function is negative on both sides of the stationary point.

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. See absolute extremum[?] for more information about this.

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.

A more straight-forward way of determining the nature of a stationary point is by examining the function values between the stationary points. However, this is limited in that it works only for functions that are continuous in at least a small interval surrounding the stationary point.



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