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:
“The most refined skills of color printing, the intricate techniques of wide-angle photography, provide us pictures of trivia bigger and more real than life. We forget that we see trivia and notice only that the reproduction is so good. Man fulfils his dream and by photographic magic produces a precise image of the Grand Canyon. The result is not that he adores nature or beauty the more. Instead he adores his camera—and himself.”
—Daniel J. Boorstin (b. 1914)
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
—Mario Vargas Llosa (b. 1936)
“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)
“The test of a vocation is the love of the drudgery it involves.”
—Logan Pearsall Smith (1865–1946)