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# Changes

→Inconclusive cases

{{maxmin test}}

==Statement==

{| class="sortable" border="1"

! 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===

* Since [[point of local extremum implies critical point]], we don't have to worry about points that are not critical points -- none of them will give local extrema.

==~~Related tests~~Facts used== # [[uses::One-sided version of second derivative test]] ==Proof==

==~~Strength of the test~~Relation with other tests==

===~~Second ~~Other tests to determine whether critical points give local extreme values=== {| class="sortable" border="1"! Test !! Quick description of how it differs from the second derivative test !! Relation with second derivative test ~~requires twice differentiability ~~|-| [[one-sided version of second derivative test]] || We compute one-sided second derivatives instead of the usual two-sided second derivative. || It is somewhat stronger and more general, i.e., it is applicable and conclusive in a somewhat wider range of circumstances.|-| [[first derivative test]] || We compute the sign of the ''first'' derivative instead of the ''second'' derivative and perform the sign computation on the ''immediate left'' and ''immediate right'' of the point rather than ''at '' the point. || It is stronger: [[second derivative test operates via first derivative test]], [[second derivative test is not stronger than first derivative test]]|-| [[higher derivative test]] || This ''begins'' with the second derivative test but ~~not around ~~computes higher derivatives ''at'' the point if the second derivative turns out to be zero, looking for the lowest order nonzero-valued derivative at the point. || It is conclusive in a somewhat wider range of circumstances. Note that the test ''begins'' the same way as the second derivative test, and we use higher derivatives only if necessary.|-| [[One-sided derivative test]] || We compute the signs of one-sided ''first'' derivatives at the point. || Not directly comparable.|} ==Requirement of twice differentiability==

The second derivative test can be applied at a critical point <math>c</math> for a function <math>f</math> only if <math>f</math> is ''twice'' differentiable at <math>c</math>. This in particular forces <math>f</math> to be once differentiable ''around'' <math>c</math>.

However, the test does ''not'' require the second derivative <math>f''</math> to be defined around <math>c</math> or to be continuous at <math>c</math>.

==~~=Relation with first derivative test=~~Ease of use==

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

The [[first derivative test]] can sometimes ''conclusively'' establish that a given critical point is not a point of local extremum. The second derivative test can ''never'' conclusively establish this. It can only conclusively establish affirmative results about local extrema.

===~~Situations when the test is inconclusive~~Inconclusive cases===

The second derivative test is inapplicable or inconclusive in the following situations. We denote the function by <math>f</math> and the critical point by <math>c</math>:

{| class="sortable" border="1"

! What problem do we run into? !! What kind of trouble do we have? !! Can we use the first derivative test? !! Link to example/explanation !! Remedy that may work (other than reverting to the first derivative test)

|-

| The critical point is of the type where <math>f'</math> is undefined || We cannot take the second derivative, because even the first derivative doesn't exist. || 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 not differentiable]] ||

|-

| <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>.

|-

| <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 ~~tests~~test]]

|}

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