Abstract The concept of maxima and minima represents one of the most fundamental applications of differential calculus, with far-reaching implications across mathematics, physics, engineering, economics, and numerous other disciplines. The theoretical underpinnings and real-world uses of derivative tests for locating and categorizing function critical points are examined in this essay. We look at the First Derivative Test, Second Derivative Test, and Higher-Order Derivative Tests, offering a thorough framework for comprehending how derivatives show how functions behave at extremal locations. Through rigorous mathematical exposition and practical examples, this study demonstrates how derivative tests serve as powerful analytical tools for optimization problems. Through dynamic and interactive graph visualization, the Desmos Graphing Calculator improves mathematics comprehension. It facilitates pupils' understanding of abstract ideas through lucid visual aids.