Point of local extremum
From Calculus
Contents
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 in the interior of the domain of a function is a point of local maximum if the following holds:
- If we are dealing with a function of one variable: There exists a value such that for all (i.e., all satisfying ).
- If we are dealing with a function of multiple variables: There exists a value such that for all satisfying the condition that the distance between and is less than .
- If we are dealing with a real-valued function on a topological space: There exists an open subset of the topological space such that , satisfying the condition that for all .
The value is termed a local maximum value.
Point of local minimum
A point in the interior of the domain of a function is a point of local minimum if the following holds:
- If we are dealing with a function of one variable: There exists a value such that for all (i.e., all satisfying ).
- If we are dealing with a function of multiple variables: There exists a value such that for all satisfying the condition that the distance between and is less than .
- If we are dealing with a real-valued function on a topological space: There exists an open subset of the topological space such that , satisfying the condition that for all .
The value 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 in the interior of the domain of a function is a point of strict local maximum if there exists a value such that for all (i.e., all satisfying ). | A point in the interior of the domain of a function is a point of strict local maximum if there exists an open subset of the domain with , such that for all . |
Point of strict local minimum | A point in the interior of the domain of a function is a point of strict local minimum if there exists a value such that for all (i.e., all satisfying ). | A point in the interior of the domain of a function is a point of strict local minimum if there exists an open subset of the domain with , such that for all . |
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 in the domain of a function is a point of local maximum from the left if there exists a value such that for all (i.e., all satisfying ). |
Point of local maximum from the right | A point in the domain of a function is a point of local maximum from the right if there exists a value such that for all (i.e., all satisfying ). |
Point of local minimum from the left | A point in the domain of a function is a point of local minimum from the left if there exists a value such that for all (i.e., all satisfying ). |
Point of local minimum from the right | A point in the domain of a function is a point of local minimum from the right if there exists a value such that for all (i.e., all satisfying ). |
Point of strict local maximum from the left | A point in the domain of a function is a point of strict local maximum from the left if there exists a value such that for all (i.e., all satisfying ). |
Point of strict local maximum from the right | A point in the domain of a function is a point of strict local maximum from the right if there exists a value such that for all (i.e., all satisfying ). |
Point of strict local minimum from the left | A point in the domain of a function is a point of strict local minimum from the left if there exists a value such that for all (i.e., all satisfying ). |
Point of strict local minimum from the right | A point in the domain of a function is a point of strict local minimum from the right if there exists a value such that for all (i.e., all satisfying ). |
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 |
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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. |