To add another item on the performance settings form you must implement the hook_for_alter or the hook_form_FORM_ID_alter.
To get the form id you just need to inspect the form an get the id attribute of the form tag:
In your custom module you will have the form alter function like this:
function mymodule_form_system_performance_settings_alter(&$form, &$form_state) {
$form['bandwidth_opt'] = array(
'#type' => 'checkbox',
'#title' => t('BANDWIDTH OPTIMIZATION'),
);
return $form;
}
Notice that the hyphens of the element id in the HTML are replaced by underscores in the function declaration.
Another good tip is the Form API reference page.
hook_form_alter
and$form['bandwidth_optimization']['my_module']