Second derivative test - Revision history

Vipul: /* Inconclusive cases */

2013-03-07T21:20:18Z

Inconclusive cases

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	<center>{{#widget:YouTube\|id=~~6dQpkFV5wt8~~}}</center>		<center>{{#widget:YouTube\|id=JbXhZSu5fnA}}</center>

Vipul: /* What the test says */

2013-03-07T21:19:21Z

What the test says

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	===Relation with critical points===		===Relation with critical points===

Vipul: /* Conclusive and inconclusive cases */

2012-05-06T23:44:38Z

Conclusive and inconclusive cases

← Older revision		Revision as of 23:44, 6 May 2012
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	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.		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~~ and ~~inconclusive~~ cases==		==Inconclusive and conclusive cases==

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

Vipul: /* Conclusive and inconclusive cases */

2012-05-06T23:42:29Z

Conclusive and inconclusive cases

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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 is inconclusive for function at critical point where second derivative is zero]] \|\| [[higher derivative test]]		\| <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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			<center>{{#widget:YouTube\|id=6dQpkFV5wt8}}</center>

Vipul: /* Statement */

2012-05-06T23:40:25Z

Statement

← Older revision		Revision as of 23:40, 6 May 2012
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			<center>{{#widget:YouTube\|id=5jPy71D8swo}}</center>
	===Relation with critical points===		===Relation with critical points===

Vipul: /* What the test says */

2012-05-05T23:27:18Z

What the test says

← Older revision		Revision as of 23:27, 5 May 2012
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	! Hypothesis on <math>f'(c)</math> !! Hypothesis on <math>f''(c)</math>!! Conclusion		! 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 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 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 local minimum 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>.		\| <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>.

Vipul: /* Proof */

2012-05-04T20:03:07Z

Proof

← Older revision		Revision as of 20:03, 4 May 2012
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	* For the other type of critical point, namely that where <math>f'</math> is undefined, the second derivative test cannot be used.		* For the other type of critical point, namely that where <math>f'</math> is undefined, the second derivative test cannot be used.
	* 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.		* 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.

			==Facts used==

			# [[uses::One-sided version of second derivative test]]

	==Proof==		==Proof==

	~~See [[second derivative test operates via first derivative test]]~~.		The proof follows directly from Fact (1).

	==Relation with other tests==		==Relation with other tests==

Vipul: /* Conclusive and inconclusive cases= */

2012-05-04T16:04:58Z

Conclusive and inconclusive cases=

← Older revision		Revision as of 16:04, 4 May 2012
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	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.		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 and inconclusive cases===		==Conclusive and inconclusive cases==

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

Vipul: /* When is the test conclusive and inconclusive? */

2012-05-04T16:04:45Z

When is the test conclusive and inconclusive?

← Older revision		Revision as of 16:04, 4 May 2012
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	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.		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.

	==~~When is the test conclusive~~ and inconclusive?==		==Conclusive and inconclusive cases===

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

Vipul: /* Situations when the test is inconclusive */

2012-05-04T16:03:45Z

Situations when the test is inconclusive

← Older revision		Revision as of 16:03, 4 May 2012
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	\| <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> 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 ~~fails~~ for function at critical point where second derivative is zero]] \|\| [[higher derivative test]]		\| <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]]
	\|}		\|}