In [5,13], stabilizing PID controllers for a class of unstable plants with time delays (I/O delays) were obtained. By utilizing this approach and methods given in [11] we aim to investigate appropriate PI, PD and PID controllers by finding the maximum allowable boundaries for each parameter. In this work, we consider a model of a flexible robot arm which includes a time delay and an integrator. We aim to find the optimal coefficients for the derivative and integral gain such that the designs achieve a set of performance and robustness objectives. Stability and robustness properties of the closed-loop system are also investigated. Specifically, while optimal derivative action gain is determined to maximize the allowable controller gain interval for PD controller, in PI controller optimal proportional gain is determined to maximize a lower bound of the largest allowable integral action gain. In other words, we utilize a method which helps us to find optimal boundaries by only investigating the proper proportional gain parameter.