You have a couple of options…
1) You need to call the destroy()
method not remove()
so…
$('#date').datepicker('destroy');
Then call your method to recreate the datepicker
object.
2) You can update the property of the existing object
via
$('#date').datepicker('option', 'minDate', new Date(startDate));
$('#date').datepicker('option', 'maxDate', new Date(endDate));
or…
$('#date').datepicker('option', { minDate: new Date(startDate),
maxDate: new Date(endDate) });