First Derivative Test - Intuitive Explanation

Intuitive Explanation

The idea behind the first derivative test is to examine the monotonic properties of a function just to the left and right of a given point in its domain. If the function "switches" from increasing to decreasing at the point, then close to that point, it will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then close to that point, it will achieve a least value at that point. If the function fails to "switch", and remains increasing or remains decreasing, then no highest or least value is achieved.

The general idea of examining monotonicity does not depend on calculus. However, calculus is introduced because there are sufficient conditions that guarantee the monotonicity properties above, and these conditions apply to the vast majority of functions one would encounter.