Critical point: Difference between revisions
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Suppose <math>f</math> is a [[function]] and <math>c</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., <math>f</math> is defined on an [[open interval]] containing <math>c</math>. | Suppose <math>f</math> is a [[function]] and <math>c</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., <math>f</math> is defined on an [[open interval]] containing <math>c</math>. | ||
Then, we say that <math>c</math> is a '''critical point''' for <math>f</math> if either the [[derivative]] <math>f'(c)</math> equals zero ''or'' <math>f</math> is not differentiable at <math>c</math> (i.e., the derivative <math>f'(c)</math> does not exist). | Then, we say that <math>c</math> is a '''critical point''' for <math>f</math> if either the [[derivative]] <math>\! f'(c)</math> equals zero ''or'' <math>f</math> is not differentiable at <math>c</math> (i.e., the derivative <math>f'(c)</math> does not exist). | ||
Note that the term ''critical point'' is ''not'' used for points at the boundary of the domain. | Note that the term ''critical point'' is ''not'' used for points at the boundary of the domain. | ||
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The term ''critical point'' is also sometimes used for the corresponding point <math>(c,f(c))</math> in the [[graph]] of <math>f</math>. | The term ''critical point'' is also sometimes used for the corresponding point <math>(c,f(c))</math> in the [[graph]] of <math>f</math>. | ||
==Facts== | |||
* [[Point of local extremum implies critical point]]: If a function attains a local extreme value (local maximum ''or'' local minimum) at a point in the interior of its domain, then that point must be a critical point. | |||
Revision as of 18:19, 20 October 2011
Definition
For a function of one variable
Suppose is a function and is a point in the interior of the domain of , i.e., is defined on an open interval containing .
Then, we say that is a critical point for if either the derivative equals zero or is not differentiable at (i.e., the derivative does not exist).
Note that the term critical point is not used for points at the boundary of the domain.
The value is termed the critical value.
The term critical point is also sometimes used for the corresponding point in the graph of .
Facts
- Point of local extremum implies critical point: If a function attains a local extreme value (local maximum or local minimum) at a point in the interior of its domain, then that point must be a critical point.