Critical point: Difference between revisions

Revision as of 18:21, 20 October 2011

Definition

For a function of one variable

Suppose $f$ is a function and $c$ is a point in the interior of the domain of $f$ , i.e., $f$ is defined on an open interval containing $c$ .

Then, we say that $c$ is a critical point for $f$ if either the derivative $f^{'} (c)$ equals zero or $f$ is not differentiable at $c$ (i.e., the derivative $f^{'} (c)$ does not exist).

Note that the term critical point is not used for points at the boundary of the domain.

The value $f (c)$ is termed the critical value.

The term critical point is also sometimes used for the corresponding point $(c, f (c))$ in the graph of $f$ .

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.

@@ Line 3: / Line 3: @@
 ===For a function of one variable===
-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).