One-sided version of higher derivative test - Revision history

Vipul at 21:34, 7 March 2013

2013-03-07T21:34:29Z

← Older revision		Revision as of 21:34, 7 March 2013
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	\| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. \|\| The first <math>k-1</matH> right hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_+(c)</math> is positive \|\| doesn't matter \|\| <math>f</math> has strict local minimum from the right at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math> or <math>f(x) := x^3, c = 0</math>		\| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. \|\| The first <math>k-1</matH> right hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_+(c)</math> is positive \|\| doesn't matter \|\| <math>f</math> has strict local minimum from the right at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math> or <math>f(x) := x^3, c = 0</math>
	\|}		\|}

			<center>{{#widget:YouTube\|id=MZ2Hj5IUWIs}}</center>

			<center>{{#widget:YouTube\|id=XvHnOUzb0SM}}</center>

	===What the test says: combined sign version with same position of first nonzero derivative on left and right===		===What the test says: combined sign version with same position of first nonzero derivative on left and right===

Vipul: /* Proof */

2012-05-04T20:48:59Z

Proof

← Older revision		Revision as of 20:48, 4 May 2012
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	We prove by induction on <math>k</math>, so we assume that the result holds for <math>k - 1</math> and then prove it for <math>k</math>.		We prove by induction on <math>k</math>, so we assume that the result holds for <math>k - 1</math> and then prove it for <math>k</math>.

	The base case of induction is Fact (1), the second derivative test.		The base case of induction is Fact (1), the one-sided version of second derivative test.

	The way we use the inductive hypothesis is that we apply it, not to the original function <math>f</math>, but to the derivative <math>f'_\pm</math>. We combine the inductive hypothesis with the fact that the derivative value is zero to conclude something about its sign on the immediate left or right, then we use Fact (2), the first derivative test.		The way we use the inductive hypothesis is that we apply it, not to the original function <math>f</math>, but to the derivative <math>f'_\pm</math>. We combine the inductive hypothesis with the fact that the derivative value is zero to conclude something about its sign on the immediate left or right, then we use Fact (2), the first derivative test.

	{{fillin}}		{{fillin}}

Vipul: /* Facts used */

2012-05-04T20:48:48Z

Facts used

← Older revision		Revision as of 20:48, 4 May 2012
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	==Facts used==		==Facts used==

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

Vipul: /* What the test says: combined sign version with same position of first nonzero derivative */

2012-05-04T20:47:55Z

What the test says: combined sign version with same position of first nonzero derivative

@@ Line 29: / Line 29: @@
 |}
-===What the test says: combined sign version with same position of first nonzero derivative===
+===What the test says: combined sign version with same position of first nonzero derivative on left and right===
 In the table below, <math>k</matH> is a positive integer greater than 1. Note that when we say <math>f</math> is differentiable a certain number of times, we mean it is differentiable at least that many times.
@@ Line 52: / Line 52: @@
 | <math>f</math> is <math>(k - 1)</math> times differentiable at and around <math>c</math>, and <math>k</math> times left and right differentiable at <math>c</math>. || All of them are equal to zero || positive || positive || odd || neither local maximum nor local minimum. The function increases through the point. || <math>f(x) := x^3, c = 0</math>
 |}

Vipul: /* What the test says: one-sided sign version */

2012-05-04T20:42:04Z

What the test says: one-sided sign version

← Older revision		Revision as of 20:42, 4 May 2012
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	! Continuity and differentiability assumption !! Assumption on one-sided derivative values <math>f^{(r)}_\pm(c), r < k</math> !! Assumption on <math>f^{(k)}_\pm(c)</math> !! Assumption on parity of <math>k</math> !! Conclusion about one-sided local extremum of <math>f</math> at <math>c</math> !! Prototypical example		! Continuity and differentiability assumption !! Assumption on one-sided derivative values <math>f^{(r)}_\pm(c), r < k</math> !! Assumption on <math>f^{(k)}_\pm(c)</math> !! Assumption on parity of <math>k</math> !! Conclusion about one-sided local extremum of <math>f</math> at <math>c</math> !! Prototypical example
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| The first <math>k-1</matH> left hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| even \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math>.<br>Here, <math>k = 2, f^{(k)}(c) = -2</math>.		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| The first <math>k-1</matH> left hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| even \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math>.<br>Here, <math>k = 2, f^{(k)}_-(c) = -2</math>.
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| The first <math>k-1</matH> left hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| odd \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := -x^3, c = 0</math>.<br>Here, <math>k = 3, f^{(k)}(c) = -6</math>.		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| The first <math>k-1</matH> left hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| odd \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := -x^3, c = 0</math>.<br>Here, <math>k = 3, f^{(k)}_-(c) = -6</math>.
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| The first <math>k-1</matH> left hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_-(c)</math> is positive \|\| even \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math>		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| The first <math>k-1</matH> left hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_-(c)</math> is positive \|\| even \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math>

Vipul: /* What the test says: one-sided sign version */

2012-05-04T20:41:35Z

What the test says: one-sided sign version

← Older revision		Revision as of 20:41, 4 May 2012
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	! Continuity and differentiability assumption !! Assumption on one-sided derivative values <math>f^{(r)}_\pm(c), r < k</math> !! Assumption on <math>f^{(k)}_\pm(c)</math> !! Assumption on parity of <math>k</math> !! Conclusion about one-sided local extremum of <math>f</math> at <math>c</math> !! Prototypical example		! Continuity and differentiability assumption !! Assumption on one-sided derivative values <math>f^{(r)}_\pm(c), r < k</math> !! Assumption on <math>f^{(k)}_\pm(c)</math> !! Assumption on parity of <math>k</math> !! Conclusion about one-sided local extremum of <math>f</math> at <math>c</math> !! Prototypical example
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| ~~All of them~~ are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| even \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math>.<br>Here, <math>k = 2, f^{(k)}(c) = -2</math>.		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| The first <math>k-1</matH> left hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| even \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math>.<br>Here, <math>k = 2, f^{(k)}(c) = -2</math>.
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| ~~All of them~~ are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| odd \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := -x^3, c = 0</math>.<br>Here, <math>k = 3, f^{(k)}(c) = -6</math>.		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| The first <math>k-1</matH> left hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| odd \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := -x^3, c = 0</math>.<br>Here, <math>k = 3, f^{(k)}(c) = -6</math>.
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| ~~All of them~~ are equal to zero \|\| <math>f^{(k)}_-(c)</math> is positive \|\| even \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math>		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| The first <math>k-1</matH> left hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_-(c)</math> is positive \|\| even \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math>
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| ~~All of them~~ are equal to zero \|\| <math>f^{(k)}_-(c)</math> is positive \|\| odd \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := x^3, c = 0</math>		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| The first <math>k-1</matH> left hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_-(c)</math> is positive \|\| odd \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := x^3, c = 0</math>
	\|-		\|-
	\| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_+(c)</math> is negative \|\| doesn't matter \|\| <math>f</math> has strict local maximum from the right at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math> or <math>f(x) := -x^3, c = 0</math>		\| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. \|\| The first <math>k-1</matH> right hand derivatives at <math>c</math>All of them are equal to zero \|\| <math>f^{(k)}_+(c)</math> is negative \|\| doesn't matter \|\| <math>f</math> has strict local maximum from the right at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math> or <math>f(x) := -x^3, c = 0</math>
	\|-		\|-
	\| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. \|\| ~~All of them~~ are equal to zero \|\| <math>f^{(k)}_+(c)</math> is positive \|\| doesn't matter \|\| <math>f</math> has strict local minimum from the right at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math> or <math>f(x) := x^3, c = 0</math>		\| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. \|\| The first <math>k-1</matH> right hand derivatives at <math>c</math> are equal to zero \|\| <math>f^{(k)}_+(c)</math> is positive \|\| doesn't matter \|\| <math>f</math> has strict local minimum from the right at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math> or <math>f(x) := x^3, c = 0</math>
	\|}		\|}

Vipul: /* What the test says: one-sided sign version */

2012-05-04T20:39:43Z

What the test says: one-sided sign version

← Older revision		Revision as of 20:39, 4 May 2012
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	! Continuity and differentiability assumption !! Assumption on one-sided derivative values <math>f^{(r)}_\pm(c), r < k</math> !! Assumption on <math>f^{(k)}_\pm(c)</math> !! Assumption on parity of <math>k</math> !! Conclusion about one-sided local extremum of <math>f</math> at <math>c</math> !! Prototypical example		! Continuity and differentiability assumption !! Assumption on one-sided derivative values <math>f^{(r)}_\pm(c), r < k</math> !! Assumption on <math>f^{(k)}_\pm(c)</math> !! Assumption on parity of <math>k</math> !! Conclusion about one-sided local extremum of <math>f</math> at <math>c</math> !! Prototypical example
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| even \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math>		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| even \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math>.<br>Here, <math>k = 2, f^{(k)}(c) = -2</math>.
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| odd \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := -x^3, c = 0</math>		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| odd \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := -x^3, c = 0</math>.<br>Here, <math>k = 3, f^{(k)}(c) = -6</math>.
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_-(c)</math> is positive \|\| even \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math>		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_-(c)</math> is positive \|\| even \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math>

Vipul: /* What the test says: one-sided sign version */

2012-05-04T20:38:43Z

What the test says: one-sided sign version

← Older revision		Revision as of 20:38, 4 May 2012
Line 16:		Line 16:
	! Continuity and differentiability assumption !! Assumption on one-sided derivative values <math>f^{(r)}_\pm(c), r < k</math> !! Assumption on <math>f^{(k)}_\pm(c)</math> !! Assumption on parity of <math>k</math> !! Conclusion about one-sided local extremum of <math>f</math> at <math>c</math> !! Prototypical example		! Continuity and differentiability assumption !! Assumption on one-sided derivative values <math>f^{(r)}_\pm(c), r < k</math> !! Assumption on <math>f^{(k)}_\pm(c)</math> !! Assumption on parity of <math>k</math> !! Conclusion about one-sided local extremum of <math>f</math> at <math>c</math> !! Prototypical example
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| negative \|\| even \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math>		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| even \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math>
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| negative \|\| odd \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := -x^3, c = 0</math>		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_-(c)</math> is negative \|\| odd \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := -x^3, c = 0</math>
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| positive \|\| even \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math>		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_-(c)</math> is positive \|\| even \|\| <math>f</math> has strict local minimum from the left at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math>
	\|-		\|-
	\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| positive \|\| odd \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := x^3, c = 0</math>		\| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_-(c)</math> is positive \|\| odd \|\| <math>f</math> has strict local maximum from the left at <math>c</math>. \|\| <math>f(x) := x^3, c = 0</math>
	\|-		\|-
	\| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. \|\| All of them are equal to zero \|\| negative \|\| doesn't matter \|\| <math>f</math> has strict local maximum from the right at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math> or <math>f(x) := -x^3, c = 0</math>		\| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_+(c)</math> is negative \|\| doesn't matter \|\| <math>f</math> has strict local maximum from the right at <math>c</math>. \|\| <math>f(x) := -x^2, c = 0</math> or <math>f(x) := -x^3, c = 0</math>
	\|-		\|-
	\| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. \|\| All of them are equal to zero \|\| positive \|\| doesn't matter \|\| <math>f</math> has strict local minimum from the right at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math> or <math>f(x) := x^3, c = 0</math>		\| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. \|\| All of them are equal to zero \|\| <math>f^{(k)}_+(c)</math> is positive \|\| doesn't matter \|\| <math>f</math> has strict local minimum from the right at <math>c</math>. \|\| <math>f(x) := x^2, c = 0</math> or <math>f(x) := x^3, c = 0</math>
	\|}		\|}

Vipul: Created page with "{{maxmin test}} {{one-sided analogue of|higher derivative test}} ==Statement== Suppose $f$ is a function of one variable and $c$ is a point in the doma..."

2012-05-04T20:37:38Z

Created page with "{{maxmin test}} {{one-sided analogue of|higher derivative test}} ==Statement== Suppose <math>f</matH> is a function of one variable and <matH>c</matH> is a point in the doma..."

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{{maxmin test}}
{{one-sided analogue of|higher derivative test}}

==Statement==

Suppose <math>f</matH> is a function of one variable and <matH>c</matH> is a point in the domain of <math>f</math>. The '''one-sided version of higher derivative test''' for <math>f</math> at <math>c</math> is a slight variation of the [[higher derivative test]] that helps determine, using one-sided higher derivatives, whether <math>f</math> has a one-sided or two-sided local extremum at <math>c</math>.

We denote by <math>f^{(r)}_-(c)</math> the function obtained by iterating the left hand derivative operation on <math>f</math> <math>r</math> times at the point <math>c</math>. Similarly, we denote by <math>f^{(r)}_+(c)</math> the function obtained by iterating the right hand derivative operation on <math>f</math> <math>r</math> times at the point <math>c</math>.

===What the test says: one-sided sign version===

In the table below, <math>k</matH> is a positive integer greater than 1.

Note that for the left side approach, the parity of <math>k</math> (even versus odd) matters for the conclusion, whereas for the right side approach, the parity of <math>k</matH> does not matter.
{| class="sortable" border="1"
! Continuity and differentiability assumption !! Assumption on one-sided derivative values <math>f^{(r)}_\pm(c), r < k</math> !! Assumption on <math>f^{(k)}_\pm(c)</math> !! Assumption on parity of <math>k</math> !! Conclusion about one-sided local extremum of <math>f</math> at <math>c</math> !! Prototypical example
|-
| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. || All of them are equal to zero || negative || even || <math>f</math> has strict local maximum from the left at <math>c</math>. || <math>f(x) := -x^2, c = 0</math>
|-
| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. || All of them are equal to zero || negative || odd || <math>f</math> has strict local minimum from the left at <math>c</math>. || <math>f(x) := -x^3, c = 0</math>
|-
| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. || All of them are equal to zero || positive || even || <math>f</math> has strict local minimum from the left at <math>c</math>. || <math>f(x) := x^2, c = 0</math>
|-
| <math>f</math> is left continuous and (at least) <math>k</math> times left differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate left of <math>c</math>. || All of them are equal to zero || positive || odd || <math>f</math> has strict local maximum from the left at <math>c</math>. || <math>f(x) := x^3, c = 0</math>
|-
| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. || All of them are equal to zero || negative || doesn't matter || <math>f</math> has strict local maximum from the right at <math>c</math>. || <math>f(x) := -x^2, c = 0</math> or <math>f(x) := -x^3, c = 0</math>
|-
| <math>f</math> is right continuous and (at least) <math>k</math> times right differentiable at <math>c</math>. Further, <math>f</math> is <math>(k - 1)</math> times (two-sided) differentiable on the immediate right of <math>c</math>. || All of them are equal to zero || positive || doesn't matter || <math>f</math> has strict local minimum from the right at <math>c</math>. || <math>f(x) := x^2, c = 0</math> or <math>f(x) := x^3, c = 0</math>
|}

===What the test says: combined sign version with same position of first nonzero derivative===

In the table below, <math>k</matH> is a positive integer greater than 1. Note that when we say <math>f</math> is differentiable a certain number of times, we mean it is differentiable at least that many times.

{| class="sortable" border="1"
! Continuity and differentiability assumption !! Assumption on derivative values <math>f^{(r)}(c), r < k</math> !! Assumption on <math>f^{(k)}_-(c)</math> !! Assumption on <math>f^{(k)}_+(c)</math> !! Assumption on parity of <math>k</math> !! Conclusion about two-sided local extremum of <math>f</math> at <math>c</math> !! Prototypical example
|-
| <math>f</math> is <math>(k - 1)</math> times differentiable at and around <math>c</math>, and <math>k</math> times left and right differentiable at <math>c</math>. || All of them are equal to zero || negative || negative || even || strict local maximum || <math>f(x) := -x^2, c = 0</math>
|-
| <math>f</math> is <math>(k - 1)</math> times differentiable at and around <math>c</math>, and <math>k</math> times left and right differentiable at <math>c</math>. || All of them are equal to zero || negative || negative || odd || neither local maximum nor local minimum. The function decreases through the point. || <math>f(x) := -x^3, c =0</math>
|-
| <math>f</math> is <math>(k - 1)</math> times differentiable at and around <math>c</math>, and <math>k</math> times left and right differentiable at <math>c</math>. || All of them are equal to zero || negative || positive || even || neither local maximum nor local minimum. The function increases through the point. || <math>f(x) := x|x|, c = 0</math>
|-
| <math>f</math> is <math>(k - 1)</math> times differentiable at and around <math>c</math>, and <math>k</math> times left and right differentiable at <math>c</math>. || All of them are equal to zero || negative || positive || odd || strict local minimum || <math>f(x) := x^2|x|, c = 0</math>
|-
| <math>f</math> is <math>(k - 1)</math> times differentiable at and around <math>c</math>, and <math>k</math> times left and right differentiable at <math>c</math>. || All of them are equal to zero || positive || negative || even || neither local maximum nor local minimum. The function decreases through the point. || <math>f(x) := -x|x|, c = 0</math>
|-
| <math>f</math> is <math>(k - 1)</math> times differentiable at and around <math>c</math>, and <math>k</math> times left and right differentiable at <math>c</math>. || All of them are equal to zero || positive || negative || odd || strict local maximum || <math>f(x) := -x^2|x|, c = 0</math>
|-
| <math>f</math> is <math>(k - 1)</math> times differentiable at and around <math>c</math>, and <math>k</math> times left and right differentiable at <math>c</math>. || All of them are equal to zero || positive || positive || even || strict local minimum || <math>f(x) := x^2, c = 0</math>
|-
| <math>f</math> is <math>(k - 1)</math> times differentiable at and around <math>c</math>, and <math>k</math> times left and right differentiable at <math>c</math>. || All of them are equal to zero || positive || positive || odd || neither local maximum nor local minimum. The function increases through the point. || <math>f(x) := x^3, c = 0</math>
|}

One-sided version of higher derivative test - Revision history

Vipul at 21:34, 7 March 2013

Vipul: /* Proof */

Vipul: /* Facts used */

Vipul: /* What the test says: combined sign version with same position of first nonzero derivative */

Vipul: /* What the test says: one-sided sign version */

Vipul: /* What the test says: one-sided sign version */

Vipul: /* What the test says: one-sided sign version */

Vipul: /* What the test says: one-sided sign version */

Vipul: Created page with "{{maxmin test}} {{one-sided analogue of|higher derivative test}} ==Statement== Suppose f is a function of one variable and c is a point in the doma..."

Vipul: Created page with "{{maxmin test}} {{one-sided analogue of|higher derivative test}} ==Statement== Suppose $f$ is a function of one variable and $c$ is a point in the doma..."