Research papers in the design and analysis of RPL objective functions investigate how different optimization criteria influence routing decisions, performance, and resource utilization in Low-Power and Lossy Networks (LLNs) that form the backbone of the Internet of Things. Since RPL (Routing Protocol for Low-Power and Lossy Networks) relies on objective functions (OFs) to guide parent selection and route construction, research focuses on designing novel OFs that balance trade-offs among energy efficiency, reliability, latency, throughput, and network lifetime. Standardized OFs such as Objective Function Zero (OF0) and the Minimum Rank with Hysteresis Objective Function (MRHOF) have been widely studied, but advanced works propose enhanced or hybrid OFs that integrate multiple metrics like Expected Transmission Count (ETX), residual energy, hop count, delay, link quality, and load balancing to meet diverse application demands. Analytical studies evaluate the mathematical models, convergence properties, and stability of different OFs, while simulation- and testbed-based works measure their effectiveness under dynamic conditions such as node mobility, high-density deployments, or bursty traffic. Researchers also investigate the security vulnerabilities of OF design, highlighting risks such as rank attacks and metric manipulation, and propose trust-aware and secure OFs that incorporate anomaly detection and lightweight cryptography. Emerging works integrate artificial intelligence and fuzzy logic techniques into OF design to enable adaptive, context-aware, and application-specific routing decisions. Application-driven studies demonstrate how tailored OFs improve performance in domains like smart healthcare, industrial IoT, precision agriculture, and intelligent transportation, where stringent requirements on delay, reliability, and scalability must be met. Collectively, this literature highlights the critical role of RPL objective functions in shaping routing efficiency, network sustainability, and resilience, while pointing to future directions in multi-metric, adaptive, and secure OF design for next-generation IoT systems.