In calculus, the second derivative test is a criterion often useful for determining whether a given critical point of a function is a local maximum or a local minimum using the value of the second derivative at the point.
The test states: If the function f is twice differentiable at a critical point x, meaning that, then:
- If then is concave down at .
- If then is concave up at .
- If, shows no concavity or is a possible inflection point.
In the last case, although the function may have a local maximum or minimum at x, because the function is sufficiently "flat" (i.e. ) the extremum is rendered undetected by the second derivative. In this case one has to examine the third derivative. The point at which is an inflection point if concavity changes on either side of it. For example, (0,0) is an inflection point on because, and and .
Read more about Second Derivative Test: Multivariable Case, Proof of The Second Derivative Test, Concavity Test
Famous quotes containing the words derivative and/or test:
“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 Adams 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 (18171862)
“There is a parallel between the twos and the tens. Tens are trying to test their abilities again, sizing up and experimenting to discover how to fit in. They dont mean everything they do and say. They are just testing. . . . Take a good deal of your daughters behavior with a grain of salt. Try to handle the really outrageous as matter-of-factly as you would a mistake in grammar or spelling.”
—Stella Chess (20th century)