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Fuzzy variational problems have received significant attention over the past decade due to a number of successful applications in fields such as optimal control theory and image segmentation. Current literature on fuzzy variational problems focuses on the necessary optimality conditions for finding the extrema, which have been studied under several differentiability conditions. In this study, we establish the sufficient conditions for the existence of minimizers for fuzzy variational problems under a weaker notion of convexity, namely preinvexity and Buckley–Feuring differentiability. We further discuss their application in a cost minimization problem.