Point of local extremum: Difference between revisions

Revision as of 18:58, 20 October 2011

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

The value $\!f(c)$ is termed a local maximum value.

Point of local minimum

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

The value $\!f(c)$ is termed a local minimum value.

Variations

Variation name	Definition
Point of strict local maximum	A point $\!c$ in the interior of the domain of a function $f$ is a point of strict local maximum if there exists a value $\delta >0$ such that $\!f(x)<f(c)$ for all $x\in (c-\delta ,c+\delta )\setminus \{c\}$ (i.e., all $x$ satisfying $0<\|x-c\|<\delta$ ).
Point of strict local minimum	A point $\!c$ in the interior of the domain of a function $f$ is a point of strict local minimum if there exists a value $\delta >0$ such that $\!f(x)>f(c)$ for all $x\in (c-\delta ,c+\delta )\setminus \{c\}$ (i.e., all $x$ satisfying $0<\|x-c\|<\delta$ ).
Point of local maximum from the left	A point $\!c$ in the domain of a function $f$ is a point of local maximum from the left if there exists a value $\delta >0$ such that $\!f(x)\leq f(c)$ for all $x\in (c-\delta ,c)$ (i.e., all $x$ satisfying $0<c-x<\delta$ ).
Point of local maximum from the right	A point $\!c$ in the domain of a function $f$ is a point of local maximum from the right if there exists a value $\delta >0$ such that $\!f(x)\leq f(c)$ for all $x\in (c,c+\delta )$ (i.e., all $x$ satisfying $0<x-c<\delta$ ).
Point of local minimum from the left	A point $\!c$ in the domain of a function $f$ is a point of local minimum from the left if there exists a value $\delta >0$ such that $\!f(x)\geq f(c)$ for all $x\in (c-\delta ,c)$ (i.e., all $x$ satisfying $0<c-x<\delta$ ).
Point of local minimum from the right	A point $\!c$ in the domain of a function $f$ is a point of local minimum from the right if there exists a value $\delta >0$ such that $\!f(x)\geq f(c)$ for all $x\in (c,c+\delta )$ (i.e., all $x$ satisfying $0<x-c<\delta$ ).
Point of strict local maximum from the left	A point $\!c$ in the domain of a function $f$ is a point of strict local maximum from the left if there exists a value $\delta >0$ such that $\!f(x)<f(c)$ for all $x\in (c-\delta ,c)$ (i.e., all $x$ satisfying $0<c-x<\delta$ ).
Point of strict local maximum from the right	A point $\!c$ in the domain of a function $f$ is a point of strict local maximum from the right if there exists a value $\delta >0$ such that $\!f(x)<f(c)$ for all $x\in (c,c+\delta )$ (i.e., all $x$ satisfying $0<x-c<\delta$ ).
Point of strict local minimum from the left	A point $\!c$ in the domain of a function $f$ is a point of strict local minimum from the left if there exists a value $\delta >0$ such that $\!f(x)>f(c)$ for all $x\in (c-\delta ,c)$ (i.e., all $x$ satisfying $0<c-x<\delta$ ).
Point of strict local minimum from the right	A point $\!c$ in the domain of a function $f$ is a point of strict local minimum from the right if there exists a value $\delta >0$ such that $\!f(x)>f(c)$ for all $x\in (c,c+\delta )$ (i.e., all $x$ satisfying $0<x-c<\delta$ ).

@@ Line 24: / Line 24: @@
 | 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>).
 |}