First derivative test for a function of multiple variables
This article describes an analogue for functions of multiple variables of the following term/fact/notion for functions of one variable: first derivative test
This article describes a test that can be used to determine whether a point in the domain of a function gives a point of local, endpoint, or absolute (global) maximum or minimum of the function, and/or to narrow down the possibilities for points where such maxima or minima occur.
View a complete list of such tests
Contents
Statement
What the test is for
The goal of the test is to determine whether a critical point for a function of multiple variables is a point where the function attains a local extreme value, and if so, whether we get a local maximum value or local minimum value.
What the test says: one-sided directional version
Suppose is a function of a vector variable and
is a point in the domain of
. Suppose
is any unit vector. Then, we have the following:
Continuity and differentiability assumptions | Hypothesis on sign of directional derivative | Conclusion for ![]() ![]() ![]() |
---|---|---|
The function ![]() ![]() ![]() ![]() ![]() ![]() |
The directional derivative ![]() ![]() |
strict local minimum |
The function ![]() ![]() ![]() ![]() ![]() ![]() |
The directional derivative ![]() ![]() |
strict local maximum |
What the test says: linear directions version
Suppose is a function of a vector variable and
is a point in the domain of
. Then, we have the following:
Continuity and differentiability assumptions | Hypothesis on sign of directional derivative | Conclusion for ![]() ![]() |
---|---|---|
![]() ![]() ![]() ![]() |
For every unit vector ![]() ![]() ![]() |
strict local minimum |
![]() ![]() ![]() ![]() |
For every unit vector ![]() ![]() ![]() |
strict local maximum |
What the test says: continuous and differentiable version
This version is weaker than the linear directions version, because it is possible to construct situations where the linear directions version is conclusive but the "continuous and differentiable" version isn't. However, in most practical situations, the two tests have the same power and same meaning.
Continuity and differentiability assumptions | Hypothesis on dot product of gradient vector and difference vector | Conclusion for ![]() ![]() |
---|---|---|
![]() ![]() ![]() ![]() |
For ![]() ![]() ![]() |
strict local minimum |
![]() ![]() ![]() ![]() |
For ![]() ![]() ![]() |
strict local maximum |