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)
“To affect the quality of the day, that is the highest of arts. Every man is tasked to make his life, even in its details, worthy of the contemplation of his most elevated and critical hour.”
—Henry David Thoreau (1817–1862)
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—Stefan Zweig (18811942)
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—Terri Apter (20th century)