Math 3
02/08
Section 2
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Concavity
Def: If the graph of f lies above all of its tangents on an interval I , then
it is called concave upward on I . If the graph of f lies below all of
tangents it is called concave down
Figure: y = x 2 Concave up
Figure: y = −x 2 Concave down
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Concavity
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Concavity Test
Recall that f 00 tell us how the derivative (slope of the tangent line)
changes with respect to x
Concavity test
1
2
If f 00 (x) > 0 for all x in I , then the graph of f is concave upward on I
If f 00 (x) < 0 for all x in I , then the graph of f is concave downward
on I
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Second derivative test
Suppose that f 00 (c) is continuous near c
1
If f 0 (c) = 0 and f 00 (c) > 0, then f has a local minimum at c.
2
If f 0 (c) = 0 and f 00 (c) < 0, then f has a local maximum at c.
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Inflection Point
A point P on a curve y = f (x) is called an inflection point if f is
continuous there and the curve changes from concave upward to concave
downward or from concave downward to concave upward at P.
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Practice
Indentify
1
Intervals where the function is increasing/decreasing.
2
Intervals where the function is concave up/concave down.
3
Mark the inflection points.
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Example 1:
f (x) = 2x 3 − 3x 2 − 36x + 4
1
Find the critical points and intervals where f (x) is
increasing/decreasing.
2
Find the intervals where f (x) is concave up/ concave down and any
inflection points of f (x).
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Example 1:
1
f (x) = 2x 3 − 3x 2 − 36x + 4
Find the critical points and intervals where f (x) is
increasing/decreasing.
Solution:
Critical points x = −2, 3. The function is
Increasing on (−∞, −2) and (3, ∞)
2
Decreasing on (−2, 3)
Find the intervals where f (x) is concave up/ concave down and any
inflection points of f (x).
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f (x) = 2x 3 − 3x 2 − 36x + 4
Example 1:
1
Find the critical points and intervals where f (x) is
increasing/decreasing.
Solution:
Critical points x = −2, 3. The function is
Increasing on (−∞, −2) and (3, ∞)
2
Decreasing on (−2, 3)
Find the intervals where f (x) is concave up/ concave down and any
inflection points of f (x).
Solution: The function is
1
Concave down (−∞, )
2
So x =
1
2
1
Concave up ( , ∞)
2
is an inflection point
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Example 1:
f (x) = 2x 3 − 3x 2 − 36x + 4
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Example 2:
f (x) = e −x (x 2 + 2x − 1)
1
Find the critical points and intervals where f (x) is
increasing/decreasing.
2
Find the intervals where f (x) is concave up/ concave down and any
inflection points of f (x).
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Example 2:
1
Find the critical points and intervals where f (x) is
increasing/decreasing.
Solution:
2
f (x) = e −x (x 2 + 2x − 1)
√ √
Critical points x = − 3, 3. The function is
√ √
Increasing on (− 3, 3)
√
√
Decreasing on (−∞, − 3) and ( 3, ∞)
Find the intervals where f (x) is concave up/ concave down and any
inflection points of f (x).
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Example 2:
1
Find the critical points and intervals where f (x) is
increasing/decreasing.
Solution:
2
f (x) = e −x (x 2 + 2x − 1)
√ √
Critical points x = − 3, 3. The function is
√ √
Increasing on (− 3, 3)
√
√
Decreasing on (−∞, − 3) and ( 3, ∞)
Find the intervals where f (x) is concave up/ concave down and any
inflection points of
Solution: The function is
Concave up (−∞, −1) and (3, ∞)
Concave down (−1, 3)
Hence x = −1, x = 3 are both inflection points
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Example 2:
Note:
√
f (x) = e −x (x 2 + 2x − 1)
3 ≈ 1.732
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