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)
“Good critical writing is measured by the perception and evaluation of the subject; bad critical writing by the necessity of maintaining the professional standing of the critic.”
—Raymond Chandler (1888–1959)
“In writing biography, fact and fiction shouldn’t be mixed. And if they are, the fictional points should be printed in red ink, the facts printed in black ink.”
—Catherine Drinker Bowen (1897–1973)
“No one can go on being a rebel too long without turning into an autocrat.”
—Lawrence Durrell (1912–1990)