$\"\"$

Definition of local extreme :

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

Definition of absolute extreme :

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 a local minimum.

$\"\"$ is a absolute maximum and it is also a local maximum.

$\"\"$ is a absolute minimum and it is also a local minimum.

$\"\"$ is a local maximum.

$\"\"$ is a local minimum.

$\"\"$

The absolute maximum is at $\"\"$ .

The absolute minimum is at $\"\"$ .

The local maximum is at $\"\"$ and $\"\"$ .

The local minimum is at $\"\"$ , $\"\"$ and $\"\"$ .