Precise Statement of First Derivative Test
The first derivative test depends on the "increasing-decreasing test", which is itself ultimately a consequence of the mean value theorem.
Suppose f is a real-valued function of a real variable defined on some interval containing the critical point x. Further suppose that f is continuous at x and differentiable on some open interval containing x, except possibly at x itself.
- If there exists a positive number r such that for every y in (x - r, x] we have f'(y) ≥ 0, and for every y in [x, x + r) we have f'(y) ≤ 0, then f has a local maximum at x.
- If there exists a positive number r such that for every y in (x - r, x) we have f'(y) ≤ 0, and for every y in (x, x + r) we have f'(y) ≥ 0, then f has a local minimum at x.
- If there exists a positive number r such that for every y in (x - r, x) ∪ (x, x + r) we have f'(y) > 0, or if there exists a positive number r such that for every y in (x - r, x) ∪ (x, x + r) we have f'(y) < 0, then f has neither a local maximum nor a local minimum at x.
- If none of the above conditions hold, then the test fails. (Such a condition is not vacuous; there are functions that satisfy none of the first three conditions.)
Again, corresponding to the comments in the section on monotonicity properites, note that in the first two cases, the inequality is not required to be strict, while in the third case, strict inequality is required.
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