We present low-temperature ultrasonic measurements on polycrystal samples of YbAuCu 4 as a function of temperature and magnetic field, which is located very near to quantum critical points. A pronounced elastic softening appears in the temperature dependence of elastic constants. Furthermore, a significant field dependence of the softening was observed, being indicative of the magnetic origin. Our results are in accordance with the deformation potential scenario for strongly enhanced quasi-particles. We argue the low temperature and elastic properties of YbAuCu 4 .