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second derivative test concavity.notebook 1 November 13, 2013 Nov 107:07 PM AP Calculus: Concavity a) Determine where f(x)=x 3 +6x 2 15x+13 is increasing and decreasing? b) What are the relative maximum and minimum of f(x)? p. 298299 # 11, 32ad, 33ac

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second derivative test concavity.notebook

1

November 13, 2013

Nov 107:07 PM

AP Calculus: Concavity

a) Determine where f(x)=x3+6x215x+13 is increasing and decreasing?

b) What are the relative maximum and minimum of f(x)?

p. 298299 # 11, 32ad, 33ac
second derivative test concavity.notebook

2

November 13, 2013

Nov 1211:44 AM
second derivative test concavity.notebook

3

November 13, 2013

Nov 107:11 PM

Concavity:

f(x) is concave down on an interval I if all of the tangents to the curve on I are above the graph of f(x)

f(x) is concave up on an interval I if all of the tangents to the curve on I are below the graph of f(x)
second derivative test concavity.notebook

4

November 13, 2013

Nov 107:16 PM

A function f(x) can be concave up, and either increasing or decreasing.

A function f(x) can be concave down, and either increasing or decreasing.
second derivative test concavity.notebook

5

November 13, 2013

Nov 107:22 PM

Where is the following function concave up? concave down?
second derivative test concavity.notebook

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November 13, 2013

Nov 107:27 PM

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 form concave downward to concave upward at P.
second derivative test concavity.notebook

7

November 13, 2013

Nov 107:26 PM

2nd Derivative test for Concavity:

If f ''(x)>0 for all x in some interval I then the graph of f is concave upward on I.

If f ''(x)
second derivative test concavity.notebook

8

November 13, 2013

Nov 107:39 PM

Using the graph of f'(x) below, find the points of inflection of f and determine where the function f is concave upward and where it is concave downward.
second derivative test concavity.notebook

9

November 13, 2013

Nov 107:31 PM

Find the points of inflection of f(x)=x3+6x215x+13 and determine the intervals where the function f is concave upward and where its is concave downward.
second derivative test concavity.notebook

10

November 13, 2013

Nov 107:32 PM

Find the points of inflection of f(x)=4x44x2 and determine the intervals where the function f is concave upward and where it is concave downward.
second derivative test concavity.notebook

11

November 13, 2013

Nov 107:39 PM

The graph of f is shown below and f is twice differentiable. Which of the following statements is true:

1.f(1)
second derivative test concavity.notebook

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November 13, 2013

Nov 1210:51 PM

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