First Derivative Test

The First Derivative Test is a method in calculus used to determine whether a critical point of a continuous function is a local maximum, local minimum, or neither. The test involves analyzing the sign of the first derivative of the function to the left and right of the critical point.

If a function f(x) has a critical point at x = c, the First Derivative Test states:

1. If f'(x) changes from positive to negative at x = c, then the function has a local maximum at x = c.
2. If f'(x) changes from negative to positive at x = c, then the function has a local minimum at x = c.
3. If f'(x) does not change sign at x = c, then the critical point is neither a local maximum nor a local minimum.

The First Derivative Test is an essential tool for finding the local extrema of a function and analyzing its behavior. This test complements other techniques in calculus, such as the Second Derivative Test and the Extreme Value Theorem.