1. Local Maximum and Minimum of a Function
   For a function , at a point within an open interval, if:
1. for all in the interval, then has a local maximum at , and is called the maximum value.
2. for all in the interval, then has a local minimum at , and is called the minimum value.
   The local maximum and minimum are collectively referred to as extremum values.


2. Extremum and Derivatives
   If a function has an extremum at and is differentiable in an open interval containing , then .
   
   However, the converse is not always true. For instance, the function has at , but it does not have an extremum at that point.
   
   Additionally, a function can have an extremum at without being differentiable at . For example, the function has a local minimum at , but it is not differentiable at .
   
   
3. Criteria for Local Maximum and Minimum
   For a differentiable function where , the behavior of around helps determine the type of extremum:
1. If changes from positive to negative around , has a local maximum at , and the maximum value is .
2. If changes from negative to positive around , has a local minimum at , and the minimum value is .