This paper considers a system with periodic inspections and periodic preventive maintenance (PM) to detect and correct hidden failures that generate a penalty cost per unit time undetected. Imperfect periodic inspections (IPIs) occur at a chosen interval $t$ , and detect hidden failures with probability $p\in (0,1)$ . Both reactive maintenance (RM), performed when a hidden failure is detected by an IPI, and PM, performed at a chosen time $(n+1)t$ , renew the system. The objective is to determine the optimal frequency $t$ and quantity $n$ of imperfect inspections between PM such that the expected cost (which includes the costs of undetected failures, IPIs, PM, and RM) per unit time is minimized over an infinite horizon. We analytically establish conditions for the existence of a finite optimal $t$ for a given value of $n$ , and discuss the asymptotic behavior of the objective function for large $n$ and $t$ . These results are further exploited to describe convergence properties of a proposed approach for finding a globally optimal solution. Also, for the special case of a Weibull time-to-failure distribution, we derive conditions that guarantee the uniqueness of a locally optimal solution for a given value of $n$ .