# Critical point for a function of multiple variables

## Contents

## Definition

### Definition in terms of gradient vector

Suppose is a function of a vector variable with coordinates . Suppose , with coordinates , is a point in the interior of the domain of . We say that is a critical point for if the gradient vector of at is either the zero vector or does not exist.

## Facts

### Relation with partial derivatives

The following is true at any point in the interior of the domain of a function:

Partial derivatives with respect to all variables are zero Critical point

The reason is as follows: the gradient vector, *if it exists*, must be the vector whose coordinates are the partial derivatives. In particular, this means that the gradient vector, *if it exists*, must be the zero vector. Thus, the gradient vector is either the zero vector or doesn't exist.

However, it's possible for a point in the domain of a function to be a critical point but to *not* have all partial derivatives equal to zero. This is because the gradient vector may fail to exist even though the partial derivatives do exist.

We can strengthen the result a bit as follows:

Partial derivatives with respect to all variables are zero or *any* of the partial derivatives is undefined Critical point

### Relation with partial derivatives and directional derivatives

We can build a little more on the previous result and state that, at any point in the interior of the domain of a function:

Directional derivatives in all linear directions are zero Partial derivatives with respect to all variables are zero Critical point

We can also say that:

Directional derivatives in all linear directions are zero or *any* of the directional derivatives is undefined Critical point