Transcript Math 71 – 1.1

Math 140
3.1 – Increasing and Decreasing Functions;
Relative Extrema
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Let 𝑓(𝑥) be defined on the interval 𝑎 < 𝑥 < 𝑏.
Suppose 𝑥1 and 𝑥2 are in that interval.
𝑓 is ___________________ on that interval if
𝑓 𝑥2 > 𝑓(𝑥1 ) when 𝑥2 > 𝑥1
𝑓 is ___________________ on that interval if
𝑓 𝑥2 < 𝑓(𝑥1 ) when 𝑥2 > 𝑥1
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Let 𝑓(𝑥) be defined on the interval 𝑎 < 𝑥 < 𝑏.
Suppose 𝑥1 and 𝑥2 are in that interval.
increasing
𝑓 is ___________________
on that interval if
𝑓 𝑥2 > 𝑓(𝑥1 ) when 𝑥2 > 𝑥1
𝑓 is ___________________ on that interval if
𝑓 𝑥2 < 𝑓(𝑥1 ) when 𝑥2 > 𝑥1
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Let 𝑓(𝑥) be defined on the interval 𝑎 < 𝑥 < 𝑏.
Suppose 𝑥1 and 𝑥2 are in that interval.
increasing
𝑓 is ___________________
on that interval if
𝑓 𝑥2 > 𝑓(𝑥1 ) when 𝑥2 > 𝑥1
decreasing
𝑓 is ___________________
on that interval if
𝑓 𝑥2 < 𝑓(𝑥1 ) when 𝑥2 > 𝑥1
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Recall: 𝑓 is increasing where 𝑓 ′ 𝑥 > 0, and
decreasing where 𝑓 ′ 𝑥 < 0.
Where might 𝑓(𝑥) change from increasing to
decreasing, or decreasing to increasing?
1. When 𝑓 ′ 𝑥 = 0
2. When 𝑓′(𝑥) does not exist (DNE)
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Recall: 𝑓 is increasing where 𝑓 ′ 𝑥 > 0, and
decreasing where 𝑓 ′ 𝑥 < 0.
Where might 𝑓(𝑥) change from increasing to
decreasing, or decreasing to increasing?
1. When 𝑓 ′ 𝑥 = 0
2. When 𝑓′(𝑥) does not exist (DNE)
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Recall: 𝑓 is increasing where 𝑓 ′ 𝑥 > 0, and
decreasing where 𝑓 ′ 𝑥 < 0.
Where might 𝑓(𝑥) change from increasing to
decreasing, or decreasing to increasing?
1. When 𝑓 ′ 𝑥 = 0
2. When 𝑓′(𝑥) does not exist (DNE)
In other words,
where might
𝑓 ′ 𝑥 change
signs?
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Recall: 𝑓 is increasing where 𝑓 ′ 𝑥 > 0, and
decreasing where 𝑓 ′ 𝑥 < 0.
Where might 𝑓(𝑥) change from increasing to
decreasing, or decreasing to increasing?
1. When 𝒇′ 𝒙 = 𝟎
2. When 𝑓′(𝑥) does not exist (DNE)
In other words,
where might
𝑓 ′ 𝑥 change
signs?
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Recall: 𝑓 is increasing where 𝑓 ′ 𝑥 > 0, and
decreasing where 𝑓 ′ 𝑥 < 0.
Where might 𝑓(𝑥) change from increasing to
decreasing, or decreasing to increasing?
1. When 𝒇′ 𝒙 = 𝟎
2. When 𝒇′(𝒙) does not exist (DNE)
In other words,
where might
𝑓 ′ 𝑥 change
signs?
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1. When 𝑓 ′ 𝑥 = 0
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1. When 𝑓 ′ 𝑥 = 0
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1. When 𝑓 ′ 𝑥 = 0
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1. When 𝑓 ′ 𝑥 = 0
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1. When 𝑓 ′ 𝑥 = 0
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2. When 𝑓′(𝑥) does not exist (DNE)
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2. When 𝑓′(𝑥) does not exist (DNE)
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2. When 𝑓′(𝑥) does not exist (DNE)
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Ex 1.
Find the intervals of increase and decrease for
𝑓 𝑥 =
𝑥2
𝑥−2
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Ex 1.
Find the intervals of increase and decrease for
𝑓 𝑥 =
𝑥2
𝑥−2
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Relative Extrema
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Relative Extrema
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Relative Extrema
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Relative Extrema
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Relative Extrema
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Relative Extrema
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If 𝑓(𝑐) is defined, and either 𝑓 ′ 𝑐 = 0 or 𝑓′(𝑐)
DNE, then 𝑥 = 𝑐 is called a ________________.
Also, 𝑐, 𝑓 𝑐
is called a __________________.
Critical numbers give us 𝑥-values where relative
extrema might occur.
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If 𝑓(𝑐) is defined, and either 𝑓 ′ 𝑐 = 0 or 𝑓′(𝑐)
DNE, then 𝑥 = 𝑐 is called a ________________.
critical number
Also, 𝑐, 𝑓 𝑐
is called a __________________.
Critical numbers give us 𝑥-values where relative
extrema might occur.
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If 𝑓(𝑐) is defined, and either 𝑓 ′ 𝑐 = 0 or 𝑓′(𝑐)
DNE, then 𝑥 = 𝑐 is called a ________________.
critical number
Also, 𝑐, 𝑓 𝑐
is called a __________________.
Critical numbers give us 𝑥-values where relative
extrema might occur.
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If 𝑓(𝑐) is defined, and either 𝑓 ′ 𝑐 = 0 or 𝑓′(𝑐)
DNE, then 𝑥 = 𝑐 is called a ________________.
critical number
Also, 𝑐, 𝑓 𝑐
critical point
is called a __________________.
Critical numbers give us 𝑥-values where relative
extrema might occur.
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If 𝑓(𝑐) is defined, and either 𝑓 ′ 𝑐 = 0 or 𝑓′(𝑐)
DNE, then 𝑥 = 𝑐 is called a ________________.
critical number
Also, 𝑐, 𝑓 𝑐
critical point
is called a __________________.
Critical numbers give us 𝑥-values where relative
extrema might occur.
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First Derivative Test for Relative Extrema
Let 𝑐 be a critical number.
𝑓′(𝑥)
+
𝑐
−
𝑥
Relative
max at
𝑐, 𝑓 𝑐
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First Derivative Test for Relative Extrema
Let 𝑐 be a critical number.
𝑓′(𝑥)
+
𝑐
Relative
max at
𝑐, 𝑓 𝑐
−
𝑥
𝑓′(𝑥)
−
𝑐
𝑥
+
Relative
min at
𝑐, 𝑓 𝑐
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First Derivative Test for Relative Extrema
Let 𝑐 be a critical number.
𝑓′(𝑥)
+
𝑐
+
𝑥
Not a relative
extremum
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First Derivative Test for Relative Extrema
Let 𝑐 be a critical number.
𝑓′(𝑥)
+
𝑐
+
Not a relative
extremum
𝑥
𝑓′(𝑥)
−
𝑐
𝑥
−
Not a relative
extremum
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Ex 2.
Find all critical numbers for
𝑓 𝑥 = 2𝑥 4 − 4𝑥 2 + 3,
and classify each critical point as a relative
maximum, relative minimum, or neither.
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Ex 2.
Find all critical numbers for
𝑓 𝑥 = 2𝑥 4 − 4𝑥 2 + 3,
and classify each critical point as a relative
maximum, relative minimum, or neither.
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Ex 3.
Find all critical numbers for 𝑓 𝑥 =
5 − 4𝑥 − 𝑥 2 ,
and classify each critical point as a relative
maximum, relative minimum, or neither.
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Ex 3.
Find all critical numbers for 𝑓 𝑥 =
5 − 4𝑥 − 𝑥 2 ,
and classify each critical point as a relative
maximum, relative minimum, or neither.
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