First Derivative Test vs Second Derivative Test for Local Extrema

When determining local extrema in calculus, two key tests are utilized: the First Derivative Test and the Second Derivative Test. The First Derivative Test examines the sign changes of the derivative around a critical point to determine if it is a local minimum or maximum. In contrast, the Second Derivative Test involves analyzing the concavity of the function at the critical point; a concave-upward shape indicates a local minimum, while a concave-downward shape indicates a local maximum. These tests are fundamental in analyzing the behavior of functions and identifying critical points in mathematical contexts.