Higher derivative test - Revision history

Vipul at 21:30, 7 March 2013

2013-03-07T21:30:57Z

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	\|}		\|}

	<center>{{#widget:YouTube\|id=~~c0MuvGV03FY~~}}</center>		<center>{{#widget:YouTube\|id=GZo5BLjzPfk}}</center>
	==Facts used==		==Facts used==

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	<center>{{#widget:YouTube\|id=~~X0DnTpdM2Rs~~}}</center>		<center>{{#widget:YouTube\|id=SKvEGy3S3cc}}</center>

	===Conclusive cases===		===Conclusive cases===
	* [[Higher derivative test is conclusive for function with algebraic derivative]]		* [[Higher derivative test is conclusive for function with algebraic derivative]]
	* [[Higher derivative test is conclusive for locally analytic function]]: In particular, it is always conclusive for polynomials, rational functions, and polynomials in <math>\sin</math> and <math>\cos</math>.		* [[Higher derivative test is conclusive for locally analytic function]]: In particular, it is always conclusive for polynomials, rational functions, and polynomials in <math>\sin</math> and <math>\cos</math>.

Vipul at 02:56, 7 May 2012

2012-05-07T02:56:24Z

← Older revision		Revision as of 02:56, 7 May 2012
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			<center>{{#widget:YouTube\|id=c0MuvGV03FY}}</center>
	==Facts used==		==Facts used==

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	\| ''All'' higher derivatives at the point are zero. \|\| The test crucially relies on determining the position and sign of the first nonzero derivative. If all higher derivatives are zero, we cannot use the test. \|\| We may or may not be able to use the first derivative test.		\| ''All'' higher derivatives at the point are zero. \|\| The test crucially relies on determining the position and sign of the first nonzero derivative. If all higher derivatives are zero, we cannot use the test. \|\| We may or may not be able to use the first derivative test.
	\|}		\|}

			<center>{{#widget:YouTube\|id=X0DnTpdM2Rs}}</center>

	===Conclusive cases===		===Conclusive cases===
	* [[Higher derivative test is conclusive for function with algebraic derivative]]		* [[Higher derivative test is conclusive for function with algebraic derivative]]
	* [[Higher derivative test is conclusive for locally analytic function]]: In particular, it is always conclusive for polynomials, rational functions, and polynomials in <math>\sin</math> and <math>\cos</math>.		* [[Higher derivative test is conclusive for locally analytic function]]: In particular, it is always conclusive for polynomials, rational functions, and polynomials in <math>\sin</math> and <math>\cos</math>.

Vipul: /* Inconclusive cases */

2012-05-06T03:36:04Z

Inconclusive cases

← Older revision		Revision as of 03:36, 6 May 2012
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	\| For critical points that are of the <math>f'(c)</math> does not exist type, the test cannot be applied. \|\| The test doesn't make sense \|\| We may or may not be able to use the first derivative test.		\| For critical points that are of the <math>f'(c)</math> does not exist type, the test cannot be applied. \|\| The test doesn't make sense \|\| We may or may not be able to use the first derivative test.
	\|-		\|-
	\| The higher derivatives cease being defined before they get a chance to become ~~zero~~. For instance, if <math>f'(c) = f''(c) = f'''(c) = 0</math> and <math>f^{(4)}(c)</math> does not exist, the test is inconclusive. \|\| The test crucially relies on determining the position and sign of the first nonzero derivative. If the higher derivative ceases to exist before it becomes nonzero, the test cannot be used. \|\| We may or may not be able to use the first derivative test.		\| The higher derivatives cease being defined before they get a chance to become nonzero. For instance, if <math>f'(c) = f''(c) = f'''(c) = 0</math> and <math>f^{(4)}(c)</math> does not exist, the test is inconclusive. \|\| The test crucially relies on determining the position and sign of the first nonzero derivative. If the higher derivative ceases to exist before it becomes nonzero, the test cannot be used. \|\| We may or may not be able to use the first derivative test.
	\|-		\|-
	\| ''All'' higher derivatives at the point are zero. \|\| The test crucially relies on determining the position and sign of the first nonzero derivative. If all higher derivatives are zero, we cannot use the test. \|\| We may or may not be able to use the first derivative test.		\| ''All'' higher derivatives at the point are zero. \|\| The test crucially relies on determining the position and sign of the first nonzero derivative. If all higher derivatives are zero, we cannot use the test. \|\| We may or may not be able to use the first derivative test.

Vipul: /* Proof */

2012-05-04T20:57:19Z

Proof

← Older revision		Revision as of 20:57, 4 May 2012
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	==Proof==		==Proof==

	The proof essentially follows from ~~the proof of~~ Fact (1).		The proof essentially follows from Fact (1), which covers a much larger body of cases.

	==Related tests==		==Related tests==

Vipul: /* What the test says */

2012-05-04T20:56:38Z

What the test says

← Older revision		Revision as of 20:56, 4 May 2012
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Case on parity of <math>k</math> !! Case on sign of <math>f^{(k)}(c)</math> !! Conclusion about <math>f</math> at <math>c</math> !! Prototypical example		! Case on parity of <math>k</math> !! Case on sign of <math>f^{(k)}(c)</math> !! Conclusion about <math>f</math> at <math>c</math> !! Prototypical example (words) !! Prototypical example (picture)
	\|-		\|-
	\| even \|\| negative \|\| strict local maximum (mimics the [[second derivative test]] where <math>k = 2</math>) \|\| <math>f(x) := -x^2, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>-x^{2n}</math> for <math>n</math> a natural number, and <math>c = 0</math>)		\| even \|\| negative \|\| strict local maximum (mimics the [[second derivative test]] where <math>k = 2</math>) \|\| <math>f(x) := -x^2, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>-x^{2n}</math> for <math>n</math> a natural number, and <math>c = 0</math>) \|\|
	\|-		\|-
	\| even \|\| positive \|\| strict local minimum (mimics the [[second derivative test]] where <math>k = 2</math>) \|\| <math>f(x) := x^2, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>x^{2n}</math> for <math>n</math> a natural number, and <math>c = 0</math>)		\| even \|\| positive \|\| strict local minimum (mimics the [[second derivative test]] where <math>k = 2</math>) \|\| <math>f(x) := x^2, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>x^{2n}</math> for <math>n</math> a natural number, and <math>c = 0</math>) \|\|
	\|-		\|-
	\| odd \|\| negative \|\| neither local maximum nor local minimum. The point is a point of decrease for the function. In fact, it is also a [[point of inflection]], though that is not of relevance here. \|\| <math>f(x) := -x^3, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>-x^{2n + 1}</math> for <math>n</math> a natural number, and <math>c = 0</math>)		\| odd \|\| negative \|\| neither local maximum nor local minimum. The point is a point of decrease for the function. In fact, it is also a [[point of inflection]], though that is not of relevance here. \|\| <math>f(x) := -x^3, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>-x^{2n + 1}</math> for <math>n</math> a natural number, and <math>c = 0</math>) \|\| [[File:Negativecube.png\|100px]]
	\|-		\|-
	\| odd \|\| positive \|\| neither local maximum nor local minimum. The point is a point of increase for the function. In fact, it is also a [[point of inflection]], though that is not of relevance here. \|\| <math>f(x) := x^3, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>x^{2n + 1}</math> for <math>n</math> a natural number, and <math>c = 0</math>)		\| odd \|\| positive \|\| neither local maximum nor local minimum. The point is a point of increase for the function. In fact, it is also a [[point of inflection]], though that is not of relevance here. \|\| <math>f(x) := x^3, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>x^{2n + 1}</math> for <math>n</math> a natural number, and <math>c = 0</math>) \|\| [[File:Cubefunctionbasic.png\|100px]]
	\|}		\|}

Vipul: /* What the test says */

2012-05-04T20:55:22Z

What the test says

← Older revision		Revision as of 20:55, 4 May 2012
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	===What the test says===		===What the test says===

	Suppose <math>f</math> is a function and <math>c</math> is a point in the interior of the [[domain]] of <math>f</math>. Suppose <math>c</math> is a [[critical point]] for <math>f</math> of the type where <math>f'(c) = 0</math>. Suppose further that there exists a natural number <math>k</~~matH~~> such that:		Suppose <math>f</math> is a function and <math>c</math> is a point in the interior of the [[domain]] of <math>f</math>. Suppose <math>c</math> is a [[critical point]] for <math>f</math> of the type where <math>f'(c) = 0</math>. Suppose further that there exists a natural number <math>k > 1</math> such that:

	* The <math>k^{th}</matH> derivative of <math>f</math> at <math>c</math>, namely <math>f^{(k)}(c)</math>, exists and is nonzero.		* The <math>k^{th}</matH> derivative of <math>f</math> at <math>c</math>, namely <math>f^{(k)}(c)</math>, exists and is nonzero.

Vipul: /* What the test says */

2012-05-04T20:54:38Z

What the test says

← Older revision		Revision as of 20:54, 4 May 2012
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Case on parity of <math>k</math> !! Case on sign of <math>f^{(k)}(c)</math> !! Conclusion about <math>f</math> at <math>c</math>		! Case on parity of <math>k</math> !! Case on sign of <math>f^{(k)}(c)</math> !! Conclusion about <math>f</math> at <math>c</math> !! Prototypical example
	\|-		\|-
	\| even \|\| negative \|\| strict local maximum (mimics the [[second derivative test]] where <math>k = 2</math>)		\| even \|\| negative \|\| strict local maximum (mimics the [[second derivative test]] where <math>k = 2</math>) \|\| <math>f(x) := -x^2, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>-x^{2n}</math> for <math>n</math> a natural number, and <math>c = 0</math>)
	\|-		\|-
	\| even \|\| positive \|\| strict local minimum (mimics the [[second derivative test]] where <math>k = 2</math>)		\| even \|\| positive \|\| strict local minimum (mimics the [[second derivative test]] where <math>k = 2</math>) \|\| <math>f(x) := x^2, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>x^{2n}</math> for <math>n</math> a natural number, and <math>c = 0</math>)
	\|-		\|-
	\| odd \|\| negative \|\| neither local maximum nor local minimum. The point is a point of decrease for the function. In fact, it is also a [[point of inflection]], though that is not of relevance here.		\| odd \|\| negative \|\| neither local maximum nor local minimum. The point is a point of decrease for the function. In fact, it is also a [[point of inflection]], though that is not of relevance here. \|\| <math>f(x) := -x^3, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>-x^{2n + 1}</math> for <math>n</math> a natural number, and <math>c = 0</math>)
	\|-		\|-
	\| odd \|\| positive \|\| neither local maximum nor local minimum. The point is a point of increase for the function. In fact, it is also a [[point of inflection]], though that is not of relevance here.		\| odd \|\| positive \|\| neither local maximum nor local minimum. The point is a point of increase for the function. In fact, it is also a [[point of inflection]], though that is not of relevance here. \|\| <math>f(x) := x^3, c = 0</math> (more generally, we can take <math>f(x)</math> as <math>x^{2n + 1}</math> for <math>n</math> a natural number, and <math>c = 0</math>)
	\|}		\|}

Vipul at 20:07, 4 May 2012

2012-05-04T20:07:02Z

← Older revision		Revision as of 20:07, 4 May 2012
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	\| odd \|\| positive \|\| neither local maximum nor local minimum. The point is a point of increase for the function. In fact, it is also a [[point of inflection]], though that is not of relevance here.		\| odd \|\| positive \|\| neither local maximum nor local minimum. The point is a point of increase for the function. In fact, it is also a [[point of inflection]], though that is not of relevance here.
	\|}		\|}

			==Facts used==

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

			==Proof==

			The proof essentially follows from the proof of Fact (1).

	==Related tests==		==Related tests==

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

2012-05-04T16:04:24Z

When is the test conclusive and inconclusive?

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	The main advantage of using the higher derivative test is that in cases where it is conclusive, the test is often easier to apply than the [[first derivative test]]. This, in turn, is because the 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 main advantage of using the higher derivative test is that in cases where it is conclusive, the test is often easier to apply than the [[first derivative test]]. This, in turn, is because the 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==

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

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

	===~~Conditions for the test to be conclusive~~===		===Conclusive cases===

	* [[Higher derivative test is conclusive for function with algebraic derivative]]		* [[Higher derivative test is conclusive for function with algebraic derivative]]
	* [[Higher derivative test is conclusive for locally analytic function]]: In particular, it is always conclusive for polynomials, rational functions, and polynomials in <math>\sin</math> and <math>\cos</math>.		* [[Higher derivative test is conclusive for locally analytic function]]: In particular, it is always conclusive for polynomials, rational functions, and polynomials in <math>\sin</math> and <math>\cos</math>.

Vipul at 16:03, 4 May 2012

2012-05-04T16:03:38Z

← Older revision		Revision as of 16:03, 4 May 2012
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	{{maxmin test}}		{{maxmin test}}
			{{remedy variation\|
			original = second derivative test\|
			defect = second derivative test is inconclusive for function at critical point where second derivative is zero}}

	==Statement==		==Statement==