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.
Common misspelling and questions (FAQ)
tationary-point sationary-point sttionary-point staionary-point statonary-point statinary-point statioary-point stationry-point stationay-point stationar-point stationarypoint stationary-oint stationary-pint stationary-pont stationary-poit stationary-poin tsationary-point sattionary-point sttaionary-point staitonary-point statoinary-point statinoary-point statioanry-point stationray-point stationayr-point stationar-ypoint stationaryp-oint stationary-opint stationary-piont stationary-ponit stationary-poitn stationary-poin sstationary-point sttationary-point staationary-point stattionary-point statiionary-point statioonary-point stationnary-point stationaary-point stationarry-point stationaryy-point stationary--point stationary-ppoint stationary-pooint stationary-poiint stationary-poinnt stationary-pointt wtationary-point atationary-point ztationary-point etationary-point xtationary-point etationary-point dtationary-point xtationary-point s5ationary-point srationary-point sfationary-point s6ationary-point sgationary-point s6ationary-point syationary-point sgationary-point stqtionary-point stwtionary-point stztionary-point stwtionary-point ststionary-point stztionary-point sta5ionary-point starionary-point stafionary-point sta6ionary-point stagionary-point sta6ionary-point stayionary-point stagionary-point stat8onary-point statuonary-point statjonary-point stat9onary-point statkonary-point stat9onary-point statoonary-point statkonary-point stati9nary-point statiinary-point statiknary-point stati0nary-point statilnary-point stati0nary-point statipnary-point statilnary-point statiohary-point statiobary-point statiojary-point statiojary-point statiomary-point stationqry-point stationwry-point stationzry-point stationwry-point stationsry-point stationzry-point stationa4y-point stationaey-point stationady-point stationa5y-point stationafy-point stationa5y-point stationaty-point stationafy-point stationar6-point stationart-point stationarg-point stationar7-point stationarh-point stationar7-point stationaru-point stationarh-point stationary0point stationaryppoint stationary[point stationary-0oint stationary-ooint stationary-loint stationary--oint stationary-;oint stationary--oint stationary-[oint stationary-;oint stationary-p9int stationary-piint stationary-pkint stationary-p0int stationary-plint stationary-p0int stationary-ppint stationary-plint stationary-po8nt stationary-pount stationary-pojnt stationary-po9nt stationary-poknt stationary-po9nt stationary-poont stationary-poknt stationary-poiht stationary-poibt stationary-poijt stationary-poijt stationary-poimt stationary-poin5 stationary-poinr stationary-poinf stationary-poin6 stationary-poing stationary-poin6 stationary-poiny stationary-poing statyonary-point statonary-point syationary-point styationary-point stationary-pointsFreed from the severity no one--absolutely his own master. And he did what too many of a jolly good time, come what might; and he had it--in his way. He and his brother-in-law, the wild young Duke of Holstein, turned gray-bearded councillors, and smashed in the windows of the staid table dishes, and broke all the benches in the palace chapel. They and ran furious races until they had ruined several fine horses. and then pelted the passers-by with sheep's heads. They spent the and ruthless boy-tricks that the period of these months of folly was young brother-in-law, who was the ringleader in all these scrapes, were--began to murmur, and when one Sunday morning three clergymen a child," the young sovereign remembered the counsels of his good and dangerous was all this reckless sport, turned over a new leaf, with the best victory of all--the conquest of himself! But though he curbed his tendency to profitless and hurtful ancestors--those rough old vikings who "despised mail and helmet manners or occupation. He hated his fair skin, and sought in every neglect of personal comfort. Many a night was passed by the boy on he slept in the hay-loft of the palace stables, without.