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:
“It is the dissenter, the theorist, the aspirant, who is quitting this ancient domain to embark on seas of adventure, who engages our interest. Omitting then for the present all notice of the stationary class, we shall find that the movement party divides itself into two classes, the actors, and the students.”
—Ralph Waldo Emerson (1803–1882)
“His misfortune was that he loved youth—he was weak to it, it kindled him. If there was one eager eye, one doubting, critical mind, one lively curiosity in a whole lecture-room full of commonplace boys and girls, he was its servant. That ardour could command him. It hadn’t worn out with years, this responsiveness, any more than the magnetic currents wear out; it had nothing to do with Time.”
—Willa Cather (1873–1947)
“the
Decapitated exclamation points in that Other Woman’s eyes.”
—Gwendolyn Brooks (b. 1917)
“As for types like my own, obscurely motivated by the conviction that our existence was worthless if we didn’t make a turning point of it, we were assigned to the humanities, to poetry, philosophy, painting—the nursery games of humankind, which had to be left behind when the age of science began. The humanities would be called upon to choose a wallpaper for the crypt, as the end drew near.”
—Saul Bellow (b. 1915)