Point of local extremum implies critical point for a function of multiple variables
From Calculus
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
This article describes an analogue for functions of multiple variables of the following term/fact/notion for functions of one variable: point of local extremum implies critical point
Contents
Statement
Suppose is a function of a vector variable . Suppose is a point in the interior of the domain of , i.e., is defined in an open ball containing .
Suppose further that is a point of local extremum for , i.e., attains a local extreme value (maximum or minimum) at .
Then the following are true:
- For every linear direction at , the directional derivative of in that direction at is either zero or it doesn't exist.
- Thus, the gradient vector of at is either zero or it doesn't exist, i.e., has a critical point at .
Facts used
- Point of local extremum implies critical point (the single variable case)
- Relation between gradient vector and directional derivatives
Proof
Directional derivative version
Given: A function of a vector variable . A point in the interior of the domain where it attains a local extreme value. A unit vector .
To prove: The directional derivative equals zero or it doesn't exist.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Consider the following function of one variable: . In other words, is the restriction of to a line through along the direction of . | ||||
2 | has a local extreme value at , which corresponds to the value . | Step (1) | Since has a local extreme value at , (a function of one variable) has a local extreme value of the same type at . | ||
3 | has a critical point at 0, i.e., or doesn't exist. | Fact (1) | Step (2) | Fact-step combination direct | |
4 | The directional derivative equals zero or it doesn't exist. | Definition of directional derivative | Step (3) | One of the definitions of directional derivative is as the ordinary derivative . Thus, this follows directly from Step (3). |
Gradient vector version
Given: A function of a vector variable . A point in the interior of the domain where it attains a local extreme value.
To prove: The gradient vector is either zero or it doesn't exist.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | For every unit vector , the directional derivative equals zero or it doesn't exist. | Directional derivative version of proof | |||
2 | If the gradient vector exists, then for every unit vector , the directional derivative exists and equals . | Fact (2) | |||
3 | If the gradient vector exists, then for every unit vector , we have | Steps (1), (2) | Step-combination direct | ||
4 | If the gradient vector exists, it must equal the zero vector. | Any vector whose dot product with every vector is zero must be the zero vector. | Step (3) | Step-fact combination direct |