Most of the power supply failure indicator circuits need a separate power supply for themselves. But the alarm circuit presented here needs no additional supply source. It employs an electrolytic capacitor to store adequate charge, to feed power to the alarm circuit which sounds an alarm for a reasonable duration when the supply fails.
Operation
Whenever the supply fails, resistor R1 pulls the base of transistor low and saturates it, turning the buzzer ON.
Schematic...