In recent years, there have been significant improvements in algorithms both for Quantified Boolean Formulas (QBF) and for Maximum Satisfiability (MaxSAT). This paper studies the problem of solving quantified formulas subject to a cost function, and considers the problem in a quantified MaxSAT setting. Two approaches are investigated. One is based on relaxing the soft clauses and performing a linear search on the cost function. The other approach, which is the main contribution of the paper, is inspired by recent work on MaxSAT, and exploits the iterative identification of unsatisfiable cores. The paper investigates the application of these approaches to the concrete problem of computing smallest minimal unsatisfiable subformulas (SMUS), a decision version of which is a well-known problem in the second level of the polynomial hierarchy. Experimental results, obtained on representative problem instances, indicate that the core-guided approach for the SMUS problem outperforms the use of linear search over the values of the cost function. More significantly, the core-guided approach also outperforms the state-of-the-art SMUS extractor Digger.