Design strategies for improving terahertz (THz) quantum cascade lasers (QCLs) in the 5-6 THz range are investigated numerically and experimentally, with the goal of overcoming the degradation in performance that occurs as the laser frequency approaches the Reststrahlen band. Two designs aimed at 5.4 THz were selected: one optimized for lower power dissipation and one optimized for better temperature performance. The active regions exhibited broadband gain, with the strongest modes lasing in the 5.3-5.6 THz range, but with other various modes observed ranging from 4.76 to 6.03 THz. Pulsed and continuous-wave (cw) operation is observed up to temperatures of 117 K and 68 K, respectively. In cw mode, the ridge laser has modes up to 5.71 THz - the highest reported frequency for a THz QCL in cw mode. The waveguide loss associated with the doped contact layers and metallization is identified as a critical limitation to performance above 5 THz.