Point of local extremum

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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 \!c in the interior of the domain of a function f is a point of local maximum if the following holds:

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

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 minimum if the following holds:

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

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

Variations

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

Variation name Definition for function of one variable General 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). A point \!c in the interior of the domain of a function f is a point of strict local maximum if there exists an open subset U of the domain with c \in U, such that \! f(x) < f(c) for all x \in U \setminus \{ c \}.
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). A point \!c in the interior of the domain of a function f is a point of strict local minimum if there exists an open subset U of the domain with c \in U, such that \! f(x) > f(c) for all x \in U \setminus \{ c \}.

Variations specific to a function of one variable

Variation name Definition for function of one variable
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) \le 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) \le 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) \ge 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) \ge 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).

Facts

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 (analogous: [SHOW MORE]) Necessary condition One-sided, but it has two-sided corollaries Appropriate one-sided differentiability One-sided derivative doesn't exist Not applicable
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
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 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 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 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.