3.3 Concavity and the SecondDerivative Test - PowerPoint PPT Presentation
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3.3 Concavity and the SecondDerivative Test
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3.3 Concavity and the Second-Derivative Test. Concavity ... Second-Derivative Test. Let f (c) = 0, and let f exist on an open interval containing c. ... – PowerPoint PPT presentation
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Title: 3.3 Concavity and the SecondDerivative Test
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3.3 Concavity and the Second-Derivative Test
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Concavity
- Let f be differentiable on an open interval I.
The graph of f is - 1) Concave upward (CU) on I if f ? is increasing
on the interval - 2) Concave downward (CD) on I if f ? is
decreasing on the interval
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Concavity
- Concave up (CU) if f ? is increasing
f ? -2
f ? 2
f ? 0
f ? is increasing
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Concavity
- Concave down (CD) I if f ? is decreasing
f ? 0
f ? -2
f ? 2
f ? is decreasing
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Intuitive Meaning of Concavity
- f is Concave upward on I if f holds coffee on
I. - f is Concave downward on I if f spills coffee on
I.
C U
C D
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Test for Concavity
- Recall f ? measures the rate of change of f with
respects to x - If f ?0 then f is increasing
- Then f ? ? measures the rate of change of f ?
with respects to x - If f ? ?0 then f ? is increasing i.e. CU
f ?
f ? 2
f ? 2
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Test for Concavity
- 1) If f ? ?(x) 0 for all x in I, then f is
concave up on I. - 2) If f ? ?(x) concave down on I.
f ?
f ? 2
f ? 2
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The Concavity Test
- Locate the x-values at which f ??(x) 0 or f
??(x) is undefined. - Use these x-values to determine the test
intervals. - Test the sign of f ??(x) in each test interval.
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Point of Inflection
- The point (c,f(c)) is a point of inflection if
the graph is - continuous at (c,f(c))
- changes from CU to CD or vice versa at (c,f(c)).
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Property of Points of Inflection
- If (c,f(c)) is a point of inflection, then
- f ??(c) 0 or f ??(c) is undefined.
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Second-Derivative Test
- Note that if f ?(c) 0 and
- If f ??(c) 0 (Concave Up), then
- f (c) is a relative minimum
- If f ??(c)
- f (c) is a relative maximum
f ?
f ? 2
f ? 2
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Second-Derivative Test
- Let f ?(c) 0, and let f ?? exist on an open
interval containing c. - If f ??(c) 0, then f(c) is a relative
minimum. - If f ??(c)
- If f ??(c) 0, then the test fails.
- Use the First-Derivative Test.
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http//www.howardcc.edu/math/MA145/3.3/3.3.htm
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