The constrained shortest path problem is a fundamental and challenging task in applications built on graphs. In this paper, we formalize and study the$Min$-$Max$resource-constrained shortest path ($Min$-$Max$RCSP) problem, which generalizes the well-studied$Max$RCSP problem. The objective is to find a simple path of minimum cost between two query nodes, subject to resource constraints between minimum and maximum limits. This problem has wide applications in fields such as delay networks and transportation. However, we theoretically prove that computing the optimal solution is NP-hard. We propose a two-stage approach that involves resource-based graph reduction followed by cost-guided path generation. To reduce the cost of expensive acyclicity checking, we introduce the technique of ancestor checking based on the shortest path tree. Furthermore, we present an even faster incremental search approach that considers both the path cost and resource constraints while avoiding acyclicity checking. Extensive experiments on twenty real graphs consistently demonstrate the superiority of our proposed methods, achieving up to two orders of magnitude improvement in time efficiency over the baseline algorithms while producing high-quality solutions.