Negative capacitance shunt damping is an effective broadband method for attenuating flexural vibration. However, proper selection of the location of the piezoelectric patches on a structure to maximize reduction has been an ongoing question in the field. Acoustic black holes are a recently developed concept to reduce vibrations on thin vibrating structures. By engineering the geometric or material properties of these thin structures, it is possible to minimize the reflected wave by gradually reducing the wave speed. However, the flexural wave speed cannot be reduced to zero on a realized structure. Therefore, when acoustic black holes are implemented, some of the incident wave energy is reflected because the wave speed must be truncated. Similarly due to the reduction in wave speed, the transverse velocity significantly increases within the acoustic black hole. It is therefore possible to add piezoelectric transducers to acoustic black hole regions on a structure to utilize negative capacitance shunt damping to address both of these issues. Consequently, the transducers are placed in the locations where the greatest control can be made and the reflected waves can be attenuated. The combination of negative capacitance shunt damping with acoustic black holes shows increased suppression of vibration over shunt damping alone.