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First Derivative Test, Concavity, Points of Inflection Section 4.3a

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Writing: True or False – A critical point of a function always signifies an extreme value of the function. Explain. FALSE!!! – Counterexample??? Do Now

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As weve seen, whether or not a critical point signifies an extreme value depends on the sign of the derivative in the immediate vicinity of the critical point: No Extreme Local Max. Local Min. Abs. Max. No Extreme

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First Derivative Test for Local Extrema The following test applies to a continuous function. At a critical point c: 1. If changes sign from positive to negative at c then has a local maximum value at c. c Local Max. c undefined

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First Derivative Test for Local Extrema The following test applies to a continuous function. At a critical point c: 2. If changes sign from negative to positive at c then has a local minimum value at c. c Local Min. c Local Min. undefined

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First Derivative Test for Local Extrema The following test applies to a continuous function. At a critical point c: 3. If does not change sign at c, then has no local extreme values at c. c No Extreme c undefined

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First Derivative Test for Local Extrema The following test applies to a continuous function. At a left endpoint a: If for x > a, then f has a local maximum (minimum) value at a. a Local Max. a Local Min.

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First Derivative Test for Local Extrema The following test applies to a continuous function. At a right endpoint b: If for x < b, then f has a local minimum (maximum) value at b. b Local Min. b Local Max.

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To use the first derivative test: Find the first derivative and any critical Find the first derivative and any critical points points Partition the x-axis into intervals using Partition the x-axis into intervals using the critical points the critical points Determine the sign of the derivative in Determine the sign of the derivative in each interval, and then use the test to each interval, and then use the test to determine the behavior of the function determine the behavior of the function

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Use the first derivative test on the given function Critical point: x = 0 (derivative undefined) Intervals Sign of Behavior of x < 0 + Increasing x > 0 + Increasing Increasing on No Extrema Can we support these answers with a graph???

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Use the first derivative test on the given function Critical points: x = 2, –2 (derivative zero) Intervals Sign of Behavior of x < –2 + Increasing –2 < x < 2 – Decreasing Increasing on and Local Max of 11 at x = –2, Local Min of –21 at x = 2 x > 2 + Increasing Decreasing on

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The cubing function is always increasing, and never decreasing… But that doesnt tell the entire story about its graph… Where on the graph of this function is the slope increasing where is it decreasing? increases decreases This leads to our definition of concavity…

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Definition: Concavity The graph of a differentiable function is (a) Concave up on an open interval I if is increasing on I. (b) Concave down on an open interval I if is decreasing on I. Concavity Test The graph of a twice-differentiable function is (a) Concave up on any interval where. (b) Concave down on any interval where. A point where the graph of a function has a tangent point of line and where the concavity changes is a point of inflection inflection.

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Lets work through #8 on p.204 CP: IP: Intervals Sign of Behavior of x < 00 < x < 11 < x < 22 < x – + + + + –– – Dec Conc up Inc Conc up Inc Conc down Dec Conc down Establish some graphical support!!!

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Lets work through #8 on p.204 (a) Increasing on (b) Decreasing on (c) Concave up on (d) Concave up on (e) Local maximum of 5 at Local minimum of –3 at (f) Inflection point

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Concavity and the Second Derivative Test

Concavity and the Second Derivative Test

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