# Difference between revisions of "Second derivative test is not stronger than first derivative test"

From Calculus

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==Statement== | ==Statement== | ||

− | It is possible to construct situations for a function and a [[critical point]] in its domain where the [[first derivative test]] is conclusive but the [[second derivative test]] is inapplicable or inconclusive. In fact, we can construct examples for every possible combination of (failure type of second derivative test) with (behavior: local max, local min, or neither). | + | It is possible to construct situations for a function and a [[critical point]] in its domain where the [[fact about::first derivative test]] is conclusive but the [[fact about::second derivative test]] is inapplicable or inconclusive. In fact, we can construct examples for every possible combination of (failure type of second derivative test) with (behavior: local max, local min, or neither). |

Note that in case the critical point is not a point of local extremum, the second derivative test is ''necessarily'' inconclusive, whereas the first derivative test may conclusively establish that the point is not a point of local extremum. | Note that in case the critical point is not a point of local extremum, the second derivative test is ''necessarily'' inconclusive, whereas the first derivative test may conclusively establish that the point is not a point of local extremum. | ||

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+ | ==Related facts== | ||

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+ | * [[Second derivative test operates via first derivative test]] |

## Latest revision as of 16:04, 2 May 2012

## Statement

It is possible to construct situations for a function and a critical point in its domain where the first derivative test is conclusive but the second derivative test is inapplicable or inconclusive. In fact, we can construct examples for every possible combination of (failure type of second derivative test) with (behavior: local max, local min, or neither).

Note that in case the critical point is not a point of local extremum, the second derivative test is *necessarily* inconclusive, whereas the first derivative test may conclusively establish that the point is not a point of local extremum.