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Second derivative test is not stronger than first derivative test

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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.

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