One-sided derivative test - Revision history

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

2013-03-07T21:28:24Z

What the test says: one-sided sign versions

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	<center>{{#widget:YouTube\|id=~~JbXhZSu5fnA~~}}</center>		<center>{{#widget:YouTube\|id=7K64X_YJUOc}}</center>

	===What the test says: combined sign versions===		===What the test says: combined sign versions===

Vipul at 21:26, 7 March 2013

2013-03-07T21:26:37Z

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	===What the test says: combined sign versions===		===What the test says: combined sign versions===

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

2012-05-07T00:09:06Z

What the test says: one-sided sign versions

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	\| <math>f</math> is right differentiable at <math>c</math> \|\| The right hand derivative of <math>f</math> at <math>c</math> is zero \|\| We cannot conclude anything		\| <math>f</math> is right differentiable at <math>c</math> \|\| The right hand derivative of <math>f</math> at <math>c</math> is zero \|\| We cannot conclude anything
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			<center>{{#widget:YouTube\|id=LvRCPawZTuo}}</center>

	===What the test says: combined sign versions===		===What the test says: combined sign versions===

Vipul: /* Facts used */

2012-05-04T21:29:21Z

Facts used

← Older revision		Revision as of 21:29, 4 May 2012
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	# [[uses::Left hand derivative is positive implies strict local maximum from the left]]		# [[uses::Left hand derivative is positive implies strict local maximum from the left]]
	# [[uses::Left hand derivative is negative implies strict local minimum from the left]]		# [[uses::Left hand derivative is negative implies strict local minimum from the left]]
	# [[uses::Right hand derivative is positive implies strict local minimum from the right]		# [[uses::Right hand derivative is positive implies strict local minimum from the right]]
	# [[uses::Right hand derivative is negative implies strict local maximum from the right]]		# [[uses::Right hand derivative is negative implies strict local maximum from the right]]

Vipul at 15:29, 4 May 2012

2012-05-04T15:29:40Z

← Older revision		Revision as of 15:29, 4 May 2012
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			{{maxmin test}}

	==Statement==		==Statement==

Vipul at 16:13, 2 May 2012

2012-05-02T16:13:03Z

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			==Facts used==

			Below are the facts for the one-sided versions:

			# [[uses::Left hand derivative is positive implies strict local maximum from the left]]
			# [[uses::Left hand derivative is negative implies strict local minimum from the left]]
			# [[uses::Right hand derivative is positive implies strict local minimum from the right]
			# [[uses::Right hand derivative is negative implies strict local maximum from the right]]

			==Proof==

			===One-sided sign versions===

			See Facts (1)-(4).

			===Combined sign versions===

			Follows directly by suitable combinations. In particular:

			* The condition for strict local maximum follows by combining (1) and (4).
			* The condition for strict local minimum follows by combining (2) and (3).
			* The condition for point of increase follows by combining (1) and (3).
			* The condition for point of decrease follows by combining (2) and (4).
	==Strength of the test==		==Strength of the test==

Vipul: Created page with "==Statement== ===What the test says: one-sided sign versions=== Suppose $f$ is a function and $c$ is a point in the domain of $f$ . We have t..."

2012-05-02T15:51:48Z

Created page with "==Statement== ===What the test says: one-sided sign versions=== Suppose <math>f</math> is a function and <math>c</math> is a point in the domain of <math>f</math>. We have t..."

New page

==Statement==

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

Suppose <math>f</math> is a function and <math>c</math> is a point in the domain of <math>f</math>. We have the following:

{| class="sortable" border="1"
! One-sided differentiability assumption !! Sign assumption on one-sided derivative !! Conclusion
|-
| <math>f</math> is left differentiable at <math>c</math> || The left hand derivative of <math>f</math> at <math>c</math> is positive || <math>f</matH> has a strict local maximum from the left at <math>c</math>
|-
| <math>f</math> is left differentiable at <math>c</math> || The left hand derivative of <math>f</math> at <math>c</math> is negative || <math>f</math> has a strict local minimum from the left at <math>c</math>
|-
| <math>f</math> is left differentiable at <math>c</math> || The left hand derivative of <math>f</math> at <math>c</math> is zero || We cannot conclude anything
|-
| <math>f</math> is right differentiable at <math>c</math> || The right hand derivative of <math>f</math> at <math>c</math> is positive || <math>f</matH> has a strict local minimum from the right at <math>c</math>
|-
| <math>f</math> is right differentiable at <math>c</math> || The right hand derivative of <math>f</math> at <math>c</math> is negative || <math>f</math> has a strict local maximum from the right at <math>c</math>
|-
| <math>f</math> is right differentiable at <math>c</math> || The right hand derivative of <math>f</math> at <math>c</math> is zero || We cannot conclude anything
|}

===What the test says: combined sign versions===

Note that in the table below, we really ''do'' need strict positivity and negativity. Note that if ''either'' one-sided derivative is zero, the test is inconclusive.

{| class="sortable" border="1"
! One-sided differentiability assumption for <math>f</math> at <math>c</math> !! Sign assumption on left hand derivative of <math>f</math> at <math>c</math> !! Sign assumption on right hand derivative of <math>f</math> at <math>c</math> !! Conclusion for <math>f</math> at <math>c</math>
|-
| both left and right differentiable (but not necessarily two-sided differentiable) || positive || negative || strict local maximum
|-
| both left and right differentiable (but not necessarily two-sided differentiable) || negative || positive || strict local minimum
|-
| both left and right differentiable (but not necessarily two-sided differentiable) || positive || positive || neither (it's a point of increase, though the function need not increase ''around'' the point)
|-
| both left and right differentiable (but not necessarily two-sided differentiable) || negative || negative || neither (it's a point of decrease, though the function need not decrease ''around'' the point)
|}

==Strength of the test==

===Relation with first derivative test===

{{further|[[first derivative test and one-sided derivative test are incomparable]]}}

The one-sided derivative test is neither strictly stronger nor strictly weaker than the first derivative test. There are situations where one test works and the other doesn't, and vice versa.

One-sided derivative test - Revision history

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

Vipul at 21:26, 7 March 2013

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

Vipul: /* Facts used */

Vipul at 15:29, 4 May 2012

Vipul at 16:13, 2 May 2012

Vipul: Created page with "==Statement== ===What the test says: one-sided sign versions=== Suppose f is a function and c is a point in the domain of f. We have t..."

Vipul: Created page with "==Statement== ===What the test says: one-sided sign versions=== Suppose $f$ is a function and $c$ is a point in the domain of $f$ . We have t..."