1. Inflection Points and the Second Derivative
Section 2.6 Day 2
Inflection Points and the Second Derivative
I can analyze the graph of f ‘’(x) in order to draw conclusions about the graph of f (x).
I can describe the graph of f ‘’(x) given f (x).
Day 4 (Answer these on your bell ringer sheet)
You are given the graph of f ’(x). For each of the graphs below, answer the following questions:
What can you say about f(x)?
What can you say about f”(x)?
a b.

2. A. For what value(s) of x is
undefined?
On what interval(s) is f(x)
concave down?.
On what intervals is
increasing?
D. On what intervals is
(-1, 1)
(-3, -1), (1, 3)
(-3, -1), (0, 1)
-1, 1
This is the graph of f(x) on (-3, 3)

3. A. For what value(s) of x is
undefined?
On what interval(s) is f(x)
concave down?.
On what intervals is
increasing?
D. On what intervals is
(-3, -1), (0, 1)
(-1, 0), (1, 3)
none
This is the graph of on (-3, 3)

4. On what interval(s) is f(x)
concave up?
List the value(s) of x for which f(x) has a point of inflection.
For what value(s) of x is ?
none
-1, 1
(-3, -1), (-1, 1), (1, 3)
This is the graph of on (-3, 3)

5. A. For what value(s) of x is
On what intervals is
f ‘ (x) > 0?
C. On what intervals is
f “ (x) < 0?
Find the x-coordinate of the
point(s) of inflection.
-0.5, 0.5
(-2, -0.5), (0.5, 2)
(-2, 0)
x = 0
This is the graph of f(x) on (-2, 2)

6. A. For what value(s) of x is
On what intervals is
f(x) decreasing?
C. On what intervals is
f “ (x) < 0?
Find the x-coordinate of the
point(s) of inflection.
-1, 0, 1
(-2, -1), (0, 1)
(-0.5, 0.5)
-0.5, 0.5
This is the graph of f ‘ (x) on
(-2, 2).

7. A. On what interval(s) is f(x)
concave up?
Find the x-coordinate of the
point(s) of inflection.
On what interval(s) is
f “ (x) > 0?
[-1, 1), (3, 5]
1, 3
This is the graph of f “ (x) on
[-1, 5].

8. 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]?
none
(-10, 0), (0, 3) -1
This is the graph of f ‘ (x)
on [-10, 3].

9. Which of the following is/are true about the function f if its
derivative is defined by
I) f is decreasing for all x < 4
II) f has a local maximum at x = 1
III) f is concave up for all 1 < x < 3
increasing NO
TRUE
A) I only B) II only C) III only D) II and III only E) I, II, and III

10. The graph of the second derivative of a function f is shown
below. Which of the following are true about the original function f?
I) The graph of f has an inflection point at x = -2
II) The graph of f has an inflection point at x = 3
III) 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 NO
YES NO

11. Which of the following statements are true about the function
f, if it’s derivative f ‘ is defined by
Use a = 2
I) The graph of f is increasing at x = 2a
II) The function f has a local maximum at x = 0
III) The graph of f has an inflection point at x = a
I only
B) I and II only
C) I and III only
D) II and III only
E) I, II and III NO