EXPLORATION OF METHODS FOR PROVING INEQUALITIES BASED ON CALCULUS THEORY

The proof of inequalities is a core subject in mathematical analysis, with methods that are both diverse and skillful. Calculus, taking functions as the object of study, provides systematic and profound methodological support for proving inequalities through core concepts such as limits, derivatives, integrals, and series expansions. This paper systematically explores typical strategies for proving inequalities using calculus tools such as function monotonicity, Mean Value Theorem, Taylor's Formula, function extrema and maximum(or minimum), and convex function theory. Combined with classic examples, it analyzes their application skills and ideological connotations, revealing the powerful tool nature and inherent logical unity of calculus in proving inequalities.