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.
Read more about this topic: First Derivative Test
Famous quotes containing the words precise, statement, derivative and/or test:
“... he held it one of the prettiest attitudes of the feminine mind to adore a man’s pre- eminence without too precise a knowledge of what it consisted in.”
—George Eliot [Mary Ann (or Marian)
“The force of truth that a statement imparts, then, its prominence among the hordes of recorded observations that I may optionally apply to my own life, depends, in addition to the sense that it is argumentatively defensible, on the sense that someone like me, and someone I like, whose voice is audible and who is at least notionally in the same room with me, does or can possibly hold it to be compellingly true.”
—Nicholson Baker (b. 1957)
“Poor John Field!—I trust he does not read this, unless he will improve by it,—thinking to live by some derivative old-country mode in this primitive new country.... With his horizon all his own, yet he a poor man, born to be poor, with his inherited Irish poverty or poor life, his Adam’s grandmother and boggy ways, not to rise in this world, he nor his posterity, till their wading webbed bog-trotting feet get talaria to their heels.”
—Henry David Thoreau (1817–1862)
“Experimental work provides the strongest evidence for scientific realism. This is not because we test hypotheses about entities. It is because entities that in principle cannot be ‘observed’ are manipulated to produce a new phenomena
[sic] and to investigate other aspects of nature.”
—Ian Hacking (b. 1936)