3,032

edits
# Changes

→Situations when the test is inconclusive

| <math>f'(c) = 0</math> but <math>f''(c)</math> is undefined || We cannot take the second derivative at the critical point. || We may or may not be able to use the first derivative test. It is possible to construct examples of (local min + conclusive first derivative test), (local max + conclusive first derivative test), (neither + conclusive first derivative test), (local min + inconclusive first derivative test), (local max + inconclusive first derivative test), (neither + inconclusive first derivative test) || [[second derivative test fails for function at critical point where it is differentiable but not twice differentiable]] || [[one-sided version of second derivative test]]: If the one-sided derivatives of <math>f'</math> exist at <math>c</math>, then we can try checking that ''both'' one-sided derivatives of <math>f'</math> have the stipulated sign for <math>f''</math>.

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| <math>f'(c) = 0</math> and <math>f''(c) = 0</math> || We are in the inconclusive case of the test as stated. || We may or may not be able to use the first derivative test. It is possible to construct examples of (local min + conclusive first derivative test), (local max + conclusive first derivative test), (neither + conclusive first derivative test), (local min + inconclusive first derivative test), (local max + inconclusive first derivative test), (neither + inconclusive first derivative test) || [[second derivative test ~~fails ~~is inconclusive for function at critical point where second derivative is zero]] || [[higher derivative test]]

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