trick questions and trick answers !
IF you increase area ( roll out the flaps ) - you can add lift and NOT appreciably add drag .
But simple flaps simply change WORKING angle of attack of the wing.
The efficient speed for this angle is lower - make the flap large enough and lower it enough and you will eventually have a "wing" that is pointing 90 degrees to direction of flight.


How can maximum lift be increased ???
If it can be increased - it was never at max.
Hows a bout this------
"an optimised airfoil is one where the shape changes to optimize lift at a given speed.
The textbook statement " flaps add lift" is simply a partial truth--
It should be " flaps add lift but at the expense of adding drag "
OR
"flaps permit a re optimized shape for lift for slower speeds, at the expense of increased drag."
It is sorta like the instruction " to go up -add up elevator "
Ir should also say " to go down, add too much up elevator".
Semantics--------