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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.
==Related facts==
* [[Second derivative test operates via first derivative test]]

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