3,033

edits
# Changes

→Inconclusive cases

! Hypothesis on <math>f'(c)</math> !! Hypothesis on <math>f''(c)</math>!! Conclusion

|-

| <math>f'(c) =0</math> || <math>f''(c) < 0</math> || <math>f</math> attains a strict local maximum value at <math>c</math> (the ''value'' is <math>f(c)</math>)

|-

| <math>f'(c) = 0</math> || <math>f''(c) > 0</math> || <math>f</math> attains a strict local minimum value at <math>c</math> (the ''value'' is <math>f(c)</math>)

|-

| <math>f'(c) = 0</math> || <math>f''(c) = 0</math> || The test is inconclusive. <math>f</math> may attain a local maximum value, a local minimum value, have a [[point of inflection]], or have some different behavior at the point <math>c</math>.

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===Relation with critical points===

The second derivative test is strictly less powerful than the [[first derivative test]], so why is it ever used? The main reason is that in cases where it is conclusive, the second derivative test is often easier to apply. This, in turn, is because the second derivative test only requires the computation of formal expressions for derivatives and evaluation of the signs of these expressions ''at a point'' rather than ''on an interval''. Evaluation at a point often requires less symbolic/algebraic manipulation.

==~~Conclusive ~~Inconclusive and ~~inconclusive ~~conclusive cases==

===The test can never be conclusive about the absence of local extrema===

| <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 is inconclusive for function at critical point where second derivative is zero]] || [[higher derivative test]]

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