If the derivative of a function changes sign around a critical point, the function is said to have a local (relative) extrema at that point. If the derivative changes from positive (increasing function) to negative (decreasing function), the function has a local (relative) maximum at the critical point. If, however, the derivative changes from negative (decreasing function) to positive (increasing function), the function has a local (relative) minimum at the critical point.
When this technique is used to determine local maximum or minimum function values, it is called the First Derivative Test for Local Extrema
. Note that there is no guarantee that the derivative will change signs, and therefore, it is essential to test each interval around a critical point.