Point of local extremum - Revision history

Vipul: /* Variations */

2015-04-02T05:12:44Z

Variations

← Older revision		Revision as of 05:12, 2 April 2015
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	The value <math>\! f(c)</math> is termed a '''local minimum value'''.		The value <math>\! f(c)</math> is termed a '''local minimum value'''.

	===Variations===		==Variations==

	===Variations applicable in all cases (does not require it to be a function of one variable)===		===Variations applicable in all cases (does not require it to be a function of one variable)===

Vipul: /* Point of local minimum */

2015-04-02T05:11:14Z

Point of local minimum

← Older revision		Revision as of 05:11, 2 April 2015
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	* '''If we are dealing with a function of multiple variables''': There exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x</math> satisfying the condition that the distance between <math>x</math> and <math>c</math> is less than <math>\delta</math>.		* '''If we are dealing with a function of multiple variables''': There exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x</math> satisfying the condition that the distance between <math>x</math> and <math>c</math> is less than <math>\delta</math>.
	* '''If we are dealing with a real-valued function on a topological space''': There exists an open subset <math>U</math> of the topological space such that <math>c \in U</math>, satisfying the condition that <math>f(x) \ge f(c)</math> for all <math>x \in U</math>.		* '''If we are dealing with a real-valued function on a topological space''': There exists an open subset <math>U</math> of the topological space such that <math>c \in U</math>, satisfying the condition that <math>f(x) \ge f(c)</math> for all <math>x \in U</math>.

	~~The value <math>\! f(c)</math> is termed a '''local minimum value'''.~~

	The value <math>\! f(c)</math> is termed a '''local minimum value'''.		The value <math>\! f(c)</math> is termed a '''local minimum value'''.

Vipul: /* Variations specific to a function of one variable */

2014-05-09T21:00:15Z

Variations specific to a function of one variable

← Older revision		Revision as of 21:00, 9 May 2014
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Variation name !! Definition for function of one variable ~~!! General definition~~		! Variation name !! Definition for function of one variable
	\|-		\|-
	\| Point of local maximum from the left \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).		\| Point of local maximum from the left \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).

Vipul: /* Variations */

2014-05-09T20:59:33Z

Variations

← Older revision		Revision as of 20:59, 9 May 2014
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	===Variations===		===Variations===

			===Variations applicable in all cases (does not require it to be a function of one variable)===
	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Variation name !! Definition		! Variation name !! Definition for function of one variable !! General definition
	\|-		\|-
	\| Point of strict local maximum \|\| A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local maximum''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) < f(c)</math> for all <math>x \in (c - \delta,c + \delta) \setminus \{ c \}</math> (i.e., all <math>x</math> satisfying <math>0 < \|x - c\| < \delta</math>).		\| Point of strict local maximum \|\| A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local maximum''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) < f(c)</math> for all <math>x \in (c - \delta,c + \delta) \setminus \{ c \}</math> (i.e., all <math>x</math> satisfying <math>0 < \|x - c\| < \delta</math>). \|\| A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local maximum''' if there exists an open subset <math>U</math> of the domain with <math>c \in U</math>, such that <math>\! f(x) < f(c)</math> for all <math>x \in U \setminus \{ c \}</math>.
	\|-		\|-
	\| Point of strict local minimum \|\| A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) > f(c)</math> for all <math>x \in (c - \delta,c + \delta) \setminus \{ c \}</math> (i.e., all <math>x</math> satisfying <math>0 < \|x - c\| < \delta</math>).		\| Point of strict local minimum \|\| A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) > f(c)</math> for all <math>x \in (c - \delta,c + \delta) \setminus \{ c \}</math> (i.e., all <math>x</math> satisfying <math>0 < \|x - c\| < \delta</math>). \|\| A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum''' if there exists an open subset <math>U</math> of the domain with <math>c \in U</math>, such that <math>\! f(x) > f(c)</math> for all <math>x \in U \setminus \{ c \}</math>.
			\|}

			===Variations specific to a function of one variable===

			{\| class="sortable" border="1"
			! Variation name !! Definition for function of one variable !! General definition
	\|-		\|-
	\| Point of local maximum from the left \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).		\| Point of local maximum from the left \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).

Vipul at 20:56, 9 May 2014

2014-05-09T20:56:37Z

← Older revision		Revision as of 20:56, 9 May 2014
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	===Point of local maximum===		===Point of local maximum===

	A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum''' if ~~there~~ exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c - \delta,c + \delta)</math> (i.e., all <math>x</math> satisfying <math>\|x - c\| < \delta</math>).		A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum''' if the following holds:

			* '''If we are dealing with a function of one variable''': There exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c - \delta,c + \delta)</math> (i.e., all <math>x</math> satisfying <math>\|x - c\| < \delta</math>).
			* '''If we are dealing with a function of multiple variables''': There exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x</math> satisfying the condition that the distance between <math>x</math> and <math>c</math> is less than <math>\delta</math>.
			* '''If we are dealing with a real-valued function on a topological space''': There exists an open subset <math>U</math> of the topological space such that <math>c \in U</math>, satisfying the condition that <math>f(x) \le f(c)</math> for all <math>x \in U</math>.

	The value <math>\! f(c)</math> is termed a '''local maximum value'''.		The value <math>\! f(c)</math> is termed a '''local maximum value'''.
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	===Point of local minimum===		===Point of local minimum===

	A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of local ~~maximum~~''' if ~~there~~ exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x \in (c - \delta,c + \delta)</math> (i.e., all <math>x</math> satisfying <math>\|x - c\| < \delta</math>).
			A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of local minimum''' if the following holds:

			* '''If we are dealing with a function of one variable''': There exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x \in (c - \delta,c + \delta)</math> (i.e., all <math>x</math> satisfying <math>\|x - c\| < \delta</math>).
			* '''If we are dealing with a function of multiple variables''': There exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x</math> satisfying the condition that the distance between <math>x</math> and <math>c</math> is less than <math>\delta</math>.
			* '''If we are dealing with a real-valued function on a topological space''': There exists an open subset <math>U</math> of the topological space such that <math>c \in U</math>, satisfying the condition that <math>f(x) \ge f(c)</math> for all <math>x \in U</math>.

			The value <math>\! f(c)</math> is termed a '''local minimum value'''.

	The value <math>\! f(c)</math> is termed a '''local minimum value'''.		The value <math>\! f(c)</math> is termed a '''local minimum value'''.

	==Variations==		===Variations===

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul: /* Facts */

2012-05-09T15:35:49Z

Facts

← Older revision		Revision as of 15:35, 9 May 2012
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	\| [[First derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| Continuous (one-sided or two-sided) at point, differentiable ''near'' the point (one-sided or two-sided), derivative has constant sign on one side (or possibly unequal constant signs on both sides) \|\| not continuous at the point, not differentiable near the point, derivative is oscillatory (in sign) near the point \|\| isolated critical points<br>always conclusive for functions with algebraic derivative, including polynomials and rational functions. Also, conclusive for locally analytic functions.		\| [[First derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| Continuous (one-sided or two-sided) at point, differentiable ''near'' the point (one-sided or two-sided), derivative has constant sign on one side (or possibly unequal constant signs on both sides) \|\| not continuous at the point, not differentiable near the point, derivative is oscillatory (in sign) near the point \|\| isolated critical points<br>always conclusive for functions with algebraic derivative, including polynomials and rational functions. Also, conclusive for locally analytic functions.
	\|-		\|-
	\| [[One-sided derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| Differentiable (one-sided or ''both'' one-sided, but not necessarily two-sided) at the point, with nonzero value of derivative ~~\|\| not one-sided differentiable, one-sided derivative is zero~~ \|\| Critical points of the one-sided derivative undefined or one-sided derivative equal to zero type \|\| not conclusive in most cases of interest to us		\| [[One-sided derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| Differentiable (one-sided or ''both'' one-sided, but not necessarily two-sided) at the point, with nonzero value of derivative \|\| Critical points of the one-sided derivative undefined or one-sided derivative equal to zero type \|\| not conclusive in most cases of interest to us
	\|-		\|-
	\| [[One-sided version of second derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| Twice differentiable (one-sided or ''both'' one-sided, but not necessarily two-sided) at the point, with first one-sided derivative zero and second one-sided derivative nonzero ~~\|\| not one-sided twice differentiable, one-sided second derivative is zero~~ \|\| Critical points of the one-sided second derivative undefined or one-sided second derivative equal to zero type \|\| not conclusive in most cases of interest to us		\| [[One-sided version of second derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| Twice differentiable (one-sided or ''both'' one-sided, but not necessarily two-sided) at the point, with first one-sided derivative zero and second one-sided derivative nonzero \|\| Critical points of the one-sided second derivative undefined or one-sided second derivative equal to zero type \|\| not conclusive in most cases of interest to us
	\|-		\|-
	\| [[Second derivative test]] \|\| Sufficient condition ''only'' \|\| Two-sided ''only'' (but has a one-sided variation) \|\| twice differentiable at the point, first derivative is zero, second derivative is nonzero ~~\|\| not twice differentiable at the point, second derivative is zero~~ \|\| Critical points that are also critical points for the derivative \|\| conclusive for functions with critical points that are only single multiplicity zeros of the derivative.		\| [[Second derivative test]] \|\| Sufficient condition ''only'' \|\| Two-sided ''only'' (but has a one-sided variation) \|\| twice differentiable at the point, first derivative is zero, second derivative is nonzero \|\| Critical points that are also critical points for the derivative \|\| conclusive for functions with critical points that are only single multiplicity zeros of the derivative.
	\|-		\|-
	\| [[One-sided version of higher derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| First derivative is zero at the point, function can be differentiated (one-sided) at the point enough times to be able to get a nonzero-valued one-sided higher derivative \|\| one-sided first derivative is undefined, one-sided derivatives cease being defined before becoming nonzero, one-sided derivatives always remain zero at the point \|\| conclusive for polynomials, rational functions, and functions with algebraic derivative, functions piecewise of this type, etc.		\| [[One-sided version of higher derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| First derivative is zero at the point, function can be differentiated (one-sided) at the point enough times to be able to get a nonzero-valued one-sided higher derivative \|\| one-sided first derivative is undefined, one-sided derivatives cease being defined before becoming nonzero, one-sided derivatives always remain zero at the point \|\| conclusive for polynomials, rational functions, and functions with algebraic derivative, functions piecewise of this type, etc.

Vipul at 15:34, 9 May 2012

2012-05-09T15:34:44Z

← Older revision		Revision as of 15:34, 9 May 2012
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	==Facts==		==Facts==

	* [[Point of local extremum ~~implies critical point]]~~		{\| class="sortable" border="1"
	* [[Critical point not implies point of local extremum]]		! Statement !! Does it establish a ''necessary'' condition for a local extremum or a ''sufficient'' condition for a local extremum? !! Is it one-sided or two-sided, or does it have versions for both? !! What are the continuity/differentiability and other assumptions for the test to be applicable and conclusive? !! Inconclusive cases !! Conclusive cases
	* [[Local maximum from the left implies left hand derivative is nonnegative if it exists]]		\|-
	* [[Local maximum from the right implies right hand derivative is nonpositive if it exists]]		\| [[Local maximum from the left implies left hand derivative is nonnegative if it exists]] (analogous: <toggledisplay>[[Local maximum from the right implies right hand derivative is nonpositive if it exists]], [[Local minimum from the left implies left hand derivative is nonpositive if it exists]], [[Local minimum from the right implies right hand derivative is nonnegative if it exists]]</toggledisplay>) \|\| Necessary condition \|\| One-sided, but it has two-sided corollaries \|\| Appropriate one-sided differentiability \|\| One-sided derivative doesn't exist \|\| Not applicable
	* [[Local minimum from the left implies left hand derivative is nonpositive if it exists]]		\|-
	* [[Local minimum from the right implies right hand derivative is nonnegative if it exists]]		\| [[Point of local extremum implies critical point]] \|\| Necessary condition \|\| Two-sided, but it follows from one-sided results \|\| None \|\| Not applicable \|\| Not applicable
			\|-
			\| [[First derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| Continuous (one-sided or two-sided) at point, differentiable ''near'' the point (one-sided or two-sided), derivative has constant sign on one side (or possibly unequal constant signs on both sides) \|\| not continuous at the point, not differentiable near the point, derivative is oscillatory (in sign) near the point \|\| isolated critical points<br>always conclusive for functions with algebraic derivative, including polynomials and rational functions. Also, conclusive for locally analytic functions.
			\|-
			\| [[One-sided derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| Differentiable (one-sided or ''both'' one-sided, but not necessarily two-sided) at the point, with nonzero value of derivative \|\| not one-sided differentiable, one-sided derivative is zero \|\| Critical points of the one-sided derivative undefined or one-sided derivative equal to zero type \|\| not conclusive in most cases of interest to us
			\|-
			\| [[One-sided version of second derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| Twice differentiable (one-sided or ''both'' one-sided, but not necessarily two-sided) at the point, with first one-sided derivative zero and second one-sided derivative nonzero \|\| not one-sided twice differentiable, one-sided second derivative is zero \|\| Critical points of the one-sided second derivative undefined or one-sided second derivative equal to zero type \|\| not conclusive in most cases of interest to us
			\|-
			\| [[Second derivative test]] \|\| Sufficient condition ''only'' \|\| Two-sided ''only'' (but has a one-sided variation) \|\| twice differentiable at the point, first derivative is zero, second derivative is nonzero \|\| not twice differentiable at the point, second derivative is zero \|\| Critical points that are also critical points for the derivative \|\| conclusive for functions with critical points that are only single multiplicity zeros of the derivative.
			\|-
			\| [[One-sided version of higher derivative test]] \|\| Sufficient condition and, in the two-sided case, necessary condition (but not a ''necessary and sufficient condition'') \|\| Both one-sided and two-sided \|\| First derivative is zero at the point, function can be differentiated (one-sided) at the point enough times to be able to get a nonzero-valued one-sided higher derivative \|\| one-sided first derivative is undefined, one-sided derivatives cease being defined before becoming nonzero, one-sided derivatives always remain zero at the point \|\| conclusive for polynomials, rational functions, and functions with algebraic derivative, functions piecewise of this type, etc.
			\|-
			\| [[Higher derivative test]] \|\| Sufficient condition and necessary condition (but not a ''necessary and sufficient condition'') \|\| two-sided (but has a one-sided version) \|\| First derivative is zero at the point, function can be differentiated (one-sided) at the point enough times to be able to get a nonzero-valued ohigher derivative \|\| first derivative is undefined, derivatives cease being defined before becoming nonzero, derivatives always remain zero at the point \|\| conclusive for nonconstant polynomials, rational functions, and functions with algebraic derivative, functions piecewise of this type, locally nonconstant locally analytic functions etc.
			\|}

Vipul: /* Variations */

2011-10-20T18:58:01Z

Variations

← Older revision		Revision as of 18:58, 20 October 2011
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	\| Point of strict local minimum \|\| A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) > f(c)</math> for all <math>x \in (c - \delta,c + \delta) \setminus \{ c \}</math> (i.e., all <math>x</math> satisfying <math>0 < \|x - c\| < \delta</math>).		\| Point of strict local minimum \|\| A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) > f(c)</math> for all <math>x \in (c - \delta,c + \delta) \setminus \{ c \}</math> (i.e., all <math>x</math> satisfying <math>0 < \|x - c\| < \delta</math>).
	\|-		\|-
	\| Point of local maximum from the left \|\| A point <math>\!c</math> in ~~the [[interior]] of~~ the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).		\| Point of local maximum from the left \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).
	\|-		\|-
	\| Point of local maximum from the right \|\| A point <math>\!c</math> in ~~the [[interior]] of~~ the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).		\| Point of local maximum from the right \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).
	\|-		\|-
	\| Point of local minimum from the left \|\| A point <math>\!c</math> in ~~the [[interior]] of~~ the [[domain]] of a [[function]] <math>f</math> is a '''point of local minimum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).		\| Point of local minimum from the left \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of local minimum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).
	\|-		\|-
	\| Point of local minimum from the right \|\| A point <math>\!c</math> in ~~the [[interior]] of~~ the [[domain]] of a [[function]] <math>f</math> is a '''point of local minimum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).		\| Point of local minimum from the right \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of local minimum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).
	\|-		\|-
	\| Point of strict local maximum from the left \|\| A point <math>\!c</math> in ~~the [[interior]] of~~ the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local maximum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) < f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).		\| Point of strict local maximum from the left \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local maximum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) < f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).
	\|-		\|-
	\| Point of strict local maximum from the right \|\| A point <math>\!c</math> in ~~the [[interior]] of~~ the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local maximum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) < f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).		\| Point of strict local maximum from the right \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local maximum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) < f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).
	\|-		\|-
	\| Point of strict local minimum from the left \|\| A point <math>\!c</math> in ~~the [[interior]] of~~ the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) > f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).		\| Point of strict local minimum from the left \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) > f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).
	\|-		\|-
	\| Point of strict local minimum from the right \|\| A point <math>\!c</math> in ~~the [[interior]] of~~ the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) > f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).		\| Point of strict local minimum from the right \|\| A point <math>\!c</math> in the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) > f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).
	\|}		\|}

Vipul: Created page with "==Definition== A '''point of local extremum''' refers to a point in the interior of the domain of a function that is either a '''point of local maximum''' or a '''po..."

2011-10-20T18:55:44Z

Created page with "==Definition== A '''point of local extremum''' refers to a point in the interior of the domain of a function that is either a '''point of local maximum''' or a '''po..."

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==Definition==

A '''point of local extremum''' refers to a point in the [[interior]] of the [[domain]] of a [[function]] that is either a '''point of local maximum''' or a '''point of local minimum'''. Both these are defined below.

===Point of local maximum===

A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c - \delta,c + \delta)</math> (i.e., all <math>x</math> satisfying <math>|x - c| < \delta</math>).

The value <math>\! f(c)</math> is termed a '''local maximum value'''.

===Point of local minimum===

A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x \in (c - \delta,c + \delta)</math> (i.e., all <math>x</math> satisfying <math>|x - c| < \delta</math>).

The value <math>\! f(c)</math> is termed a '''local minimum value'''.

==Variations==

{| class="sortable" border="1"
! Variation name !! Definition
|-
| Point of strict local maximum || A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local maximum''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) < f(c)</math> for all <math>x \in (c - \delta,c + \delta) \setminus \{ c \}</math> (i.e., all <math>x</math> satisfying <math>0 < |x - c| < \delta</math>).
|-
| Point of strict local minimum || A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) > f(c)</math> for all <math>x \in (c - \delta,c + \delta) \setminus \{ c \}</math> (i.e., all <math>x</math> satisfying <math>0 < |x - c| < \delta</math>).
|-
| Point of local maximum from the left || A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).
|-
| Point of local maximum from the right || A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of local maximum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \le f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).
|-
| Point of local minimum from the left || A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of local minimum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).
|-
| Point of local minimum from the right || A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of local minimum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) \ge f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).
|-
| Point of strict local maximum from the left || A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local maximum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) < f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).
|-
| Point of strict local maximum from the right || A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local maximum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) < f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).
|-
| Point of strict local minimum from the left || A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum from the left''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) > f(c)</math> for all <math>x \in (c - \delta,c)</math> (i.e., all <math>x</math> satisfying <math>0 < c - x < \delta</math>).
|-
| Point of strict local minimum from the right || A point <math>\!c</math> in the [[interior]] of the [[domain]] of a [[function]] <math>f</math> is a '''point of strict local minimum from the right''' if there exists a value <math>\delta > 0</math> such that <math>\! f(x) > f(c)</math> for all <math>x \in (c, c + \delta)</math> (i.e., all <math>x</math> satisfying <math>0 < x - c < \delta</math>).
|}

==Facts==

* [[Point of local extremum implies critical point]]
* [[Critical point not implies point of local extremum]]
* [[Local maximum from the left implies left hand derivative is nonnegative if it exists]]
* [[Local maximum from the right implies right hand derivative is nonpositive if it exists]]
* [[Local minimum from the left implies left hand derivative is nonpositive if it exists]]
* [[Local minimum from the right implies right hand derivative is nonnegative if it exists]]