Section 2.6 Day 2Inflection Points and the Second Derivative

I can describe the graph of f ‘’(x) given f (x).

I can analyze the graph of f ‘’(x) in order to draw conclusions about the graph of f (x).

You are given the graph of f ’(x). For each of the graphs below, answer the following questions:

1. What can you say about f(x)?

2. What can you say about f”(x)?

a. b.

Day 4 (Answer these on your bell ringer sheet)

A. For what value(s) of x is undefined? B.On what interval(s) is f(x) concave down?.

 C.On what intervals is increasing?  

D. On what intervals is

This is the graph of f(x) on (-3, 3)

(-1, 1)

(-3, -1), (1, 3)

(-3, -1), (0, 1)

-1, 1

f ' x

f ' x

f ' x 0?

A. For what value(s) of x is undefined? B.On what interval(s) is f(x) concave down?.

 C.On what intervals is increasing?  

D. On what intervals is

This is the graph of on (-3, 3)

(-3, -1), (0, 1)

(-1, 0), (1, 3)

none

none

f ' x

f ' x

f ' x 0? f ' x

A. On what interval(s) is f(x) concave up? 

B. List the value(s) of x for which f(x) has a point of inflection.

C. For what value(s) of x is ?  

This is the graph of on (-3, 3)

none

-1, 1

(-3, -1), (-1, 1), (1, 3)

f " x

f " x 0

A. For what value(s) of x is f ‘ (x) = 0? 

B.On what intervals is f ‘ (x) > 0?

 C. On what intervals is f “ (x) < 0?

 D.Find the x-coordinate of the point(s) of inflection. This is the graph of f(x) on (-2, 2)

-0.5, 0.5

(-2, -0.5), (0.5, 2)

(-2, 0)

x = 0

A. For what value(s) of x is f ‘ (x) = 0? 

B.On what intervals is f(x) decreasing?

 C. On what intervals is f “ (x) < 0?

 D.Find the x-coordinate of the point(s) of inflection. This is the graph of f ‘ (x) on

(-2, 2).

-1, 0, 1

(-2, -1), (0, 1)

(-0.5, 0.5)

-0.5, 0.5

A. On what interval(s) is f(x) concave up? 

B.Find the x-coordinate of the point(s) of inflection.  

C.On what interval(s) is f “ (x) > 0? 

This is the graph of f “ (x) on [-1, 5].

[-1, 1), (3, 5]

1, 3

[-1, 1), (3, 5]

For what value(s) of x does f ‘ (x) not exist?

 On what interval(s) is f(x) concave down? 

On what interval(s) is f “ (x) > 0?

 Where is/are the relative minima on [-10, 3]?This is the graph of f ‘ (x)

on [-10, 3].

none

none

(-10, 0), (0, 3)

-1

Which of the following is/are true about the function f if itsderivative is defined by 2

f ' x x 1 4 x ? I) f is decreasing for all x < 4II) f has a local maximum at x = 1III) f is concave up for all 1 < x < 3

A) I only B) II only C) III only D) II and III only E) I, II, and III

increasingNO

TRUE

The graph of the second derivative of a function f is shown below. Which of the following are true about the original functionf? I) The graph of f has an inflection point at x = -2 II) The graph of f has an inflection point at x = 3III) The graph of f is concave down on the interval (0, 4)

A) I only B) II only C) III only D) I and II only E) I, II and III

NOYESNO

Which of the following statements are true about the functionf, if it’s derivative f ‘ is defined by 3

f ' x x x a , a 0?

I) The graph of f is increasing at x = 2a II) The function f has a local maximum at x = 0III) The graph of f has an inflection point at x = a

A) I only B) I and II only C) I and III only D) II and III onlyE) I, II and III

3I) f ' 4 4 4 2 0 YES

3II) f ' 0 0 0 2 0 YES

3III) f ' x x x 2 Graph f " x NO

Use a = 2