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Geometry of Functions IX: Concavity
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Lesson Summary:
In this lesson we define the concept of concavity and show in pictures when a function is concave up or concave down.
Lesson Inputs:
Geometry of Functions VIII: The Second Derivative Test
Lesson Outputs:
Geometry of Functions XI: Inflection Points
Geometry of Functions X: Determining Concavity From Second Derivatives
GAIN AN ADVANTAGE
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Lesson Specific Problems
Prove
f
(
x
)
=
1
x
is concave down on the open interval
(
−
∞
,
0
)
and concave up on the open interval
(
0
,
∞
)
.
Prove that
f
(
x
)
=
x
has no concavity.
Prove
x
3
is concave down on the interval
(
−
∞
,
0
)
and concave up on the interval
(
0
,
∞
)
.
Prove that
f
(
x
)
=
x
2
is concave up.