Cultivating optimization intuition in engineering education through multi-methodological problem solving

This paper presents a m.ulti-methodological approach to solving optimization problems within the framework of engineering education. While traditional calculus-based methods focusing on derivatives are the standard pedagogical tool, they often fail to provide students with a deep physical and geometric intuition. We analyze several complex optimization cases, including radical sums, structural trapezoidal geometry, and fundamental mathematical constants, to demonstrate how visual-geometric methods can yield more efficient and insightful solutions. The results suggest that integrating these "short-cut" geometric strategies alongside traditional calculus enhances the analytical capabilities of engineering students and fosters a more robust understanding of mathematical modeling in practical applications.