StellarThe First Derivative Test uses sign changes in f'(x) to determine whether a critical point is a local maximum, a local minimum, or neither. If f'(x) changes from positive to negative at a critical point c, then f changes from increasing to decreasing, so f(c) is a local maximum. If f'(x) changes from negative to positive, then f changes from decreasing to increasing, so f(c) is a local minimum. If the derivative does not change sign, then the critical point is not a local extremum. This test gives a direct connection between derivative sign and function shape.
The First Derivative Test is powerful because it uses actual behavior on both sides of a critical point rather than only the value of the derivative at the point itself. It helps students avoid the mistake of assuming every critical point is an extremum. The process is: find critical points, make a sign chart for f'(x), then interpret the sign changes. This method is one of the main tools for analyzing graphs in AP Calculus.