Stationary Points, Critical Points and Turning Points
The term "critical point" may be confused with "stationary point". Critical point is more general: a critical point is either a stationary point or a point where the derivative is not defined. Thus a stationary point is always a critical point, but a critical point is not always a stationary point: it might also be a non-differentiable point. For a smooth function, these are interchangeable, hence the confusion; when a function is understood to be smooth, one can refer to stationary points as critical points, but when it may be non-differentiable, one should distinguish these notions. There is also a different definition of critical point in higher dimensions, where the derivative does not have full rank, but is not necessarily zero; this is not analogous to stationary points, as the function may still be changing in some direction.
A turning point is a point at which the derivative changes sign. A turning point may be either a local maximum or a minimum point. If the function is smooth, then the turning point must be a stationary point, however not all stationary points are turning points, for example has a stationary point at x=0, but the derivative doesn't change sign as there is a point of inflexion at x=0.
Read more about this topic: Stationary Point
Famous quotes containing the words stationary, critical, points and/or turning:
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—Ralph Waldo Emerson (1803–1882)
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—Wyndham Lewis (1882–1957)
“It’s my feeling that God lends you your children until they’re about eighteen years old. If you haven’t made your points with them by then, it’s too late.”
—Betty Ford (b. 1918)
“It was a maxim with Mr. Brass that the habit of paying compliments kept a man’s tongue oiled without any expense; and that, as that useful member ought never to grow rusty or creak in turning on its hinges in the case of a practitioner of the law, in whom it should be always glib and easy, he lost few opportunities of improving himself by the utterance of handsome speeches and eulogistic expressions”
—Charles Dickens (1812–1870)