The First Derivative Test
The First Derivative Test (Motivation and Theorem)
If f is a function, then f has a relative maximum at x = c if for all points a near c, f(c) > f(a), and f has a relative minimum at x = c if for all points a near c, f(c) < f(a).
Consider a relative maximum, we have that on the left, the function is increasing and on the right the function is decreasing. Similarly, for a relative minimum, on the right the function is decreasing and on the left the function is increasing.
We can now state the first derivative test:
The First Derivative Test
Let f be a differentiable function with f (c) = 0 then
1. If f (x) changes from positive to negative, then f has a relative maximum at c.
2. If f (x) changes from negative to positive, then f has a relative minimum at c.