In the intracellular signaling networks that regulate important cell processes, the base pattern comprises the cycle of reversible phosphorylation of a protein, catalyzed by kinases and opposing phosphatases. Mathematical modeling and analysis have been used for gaining a better understanding of their functions and to capture the rules governing system behavior. Since biochemical parameters in signaling pathways are not easily accessible experimentally, it is necessary to explore possibilities for both steady-state and dynamic responses in these systems. While a number of studies have focused on analyzing these properties separately, it is necessary to take into account both of these responses simultaneously in order to be able to interpret a broader range of phenotypes. This paper investigates the trade-offs between optimal characteristics of both steady-state and dynamic responses. Following an inverse sensitivity analysis approach, we use systematic optimization methods to find the biochemical and biophysical parameters that simultaneously achieve optimal steady-state and dynamic performance. Remarkably, we find that even a single covalent modification cycle can simultaneously and robustly achieve high ultrasensitivity, high amplification and rapid signal transduction. We also find that the response rise and decay times can be modulated independently by varying the activating- and deactivating-enzyme-to-interconvertible-protein ratios.