Definition of local extrema :

Functions can have "hills and valleys"  places where they reach a minimum or maximum value.

Definition of absolute extrema :

The maximum or minimum over the entire function is called an "Absolute" or "Global" maximum or minimum.

There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum.

Observe the graph.

 is absolute minimum, but  is not a local minima, because it is an end point.

 is local maximum, because moving from left to right or vice versa,  increases.

 is none, because moving to left  increases and moving to right  decreases.

Since  on its domain ,  is absolute maximum.

 is local minimum, because moving from left to right,  increases.

 is none, because moving to left  decreases.

Absolute maximum at .

Absolute minimum at .

Local maximum at .

Local Minimum .

Neither maximum nor minimum at  and .