Section 3 - Ms. Monaco's Math...

Section 3.4

Concavity and the Second Derivative Test

Concavity

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Definition of Concavity: f is concave upward when f’ is increasing and f is concave downward when f’ is decreasing

Test for Concavity

• Let f be a function whose second derivative exits on an open interval I.

1. If for all x in I, then the graph of f is concave upward in I.

2. If for all x in I, then the graph of f is concave downward in I.

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0)( xf

Definition of a Point of Inflection

• A point of inflection is an ordered pair where a graph changes concavity.

• THEOREM:

If (c, f(c)) is a point of inflection of the graph of f, then either or is not differentiable at

x = c.

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Draw a function with an inflection point

• Ex1: a) Determine the open intervals on which the graph is concave upward or downward and state any points of inflection

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xxf

• Ex1: b) Determine the open intervals on which the graph is concave upward or downward and state any points of inflection

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The Second Derivative Test

• This test is an alternative way to find the relative maximums and minimums of a function

• If , the test fails and you must use the fist derivative test

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Ex2: Find the relative extrema for

using the second derivative test.

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Homework

• p. 195 # 1-3 odd, 13-49 EOO, 61-65 odd