Definition of Relative Extrema :

1. If there is an open interval containing  on which  is a maximum, then  is called a relative maximum of  or you can say that  has a relative maximum at .

2. If there is an open interval containing  on which  is a minimum, then  is called a relative minimum of  or you can say that  has a relative minimum at .

Definition of Global Extrema :

The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum, or the global minimum and global maximum.

Now observe the graph :

The graph increasing over interval .

The graph is decreasing over the interval  .

So the function has relative maximum at .

Relative maximum at .

Again the graph increasing over interval .

The function has relative minimum at .

Relative minimum at .

The graph is decreasing over the interval  .

The function has relative maximum at .

Relative maximum at .

The function has relative maximum at  and  in the entire interval that is .

Therefore relative maximum is a absolute maximum.

The graph has relative and absolute at  and .

The graph has only one relative at , so it is the absolute minimum.

If  has a relative minimum or relative maximum at  then  is a critical number of .

So the function has a critical numbers at ,  and .

The function has a critical numbers at ,  and .

The graph has relative and absolute maximum at  and .

The graph has relative and absolute minimum at . \ \