1. Concavity of a Function
Unit1– Derivative Graphs
Section 11.2 – Second Derivative Graphs
Second Derivative
Concavity of a Function

2. f “ (a) gives us information on the concavity of the graph of f(x).
If f ”(a) > 0, the graph of f(x) is concave up
If f ”(a) < 0, the graph of f(x) is concave down

3. If f ” (a) = 0, a point of inflection MAY exist on f.
A point of inflection occurs when concavity changes

4. The graph is on the interval (-1, 5)
Given f(x), find the x-coordinates of the point(s) of inflection on f 2
B. Given f ‘ (x) is graphed, find the x-coordinates of the point(s) of inflection on f.
1, 3
C. If f(x) is graphed, for what intervals is f ‘ ‘ (x) negative?
(-1, 2)
D. If f ’ (x) is graphed, for what intervals is f “ (x) positive?
(-1, 1) and (3, 5)
The graph is on the interval (-1, 5)
E. If f “ (x) is graphed, for what intervals is f “ (x) negative?
(-1, 0)

5. The graph is on the interval [-2, 2]
Given f(x), at what values of x are the points of inflection?
Given f ‘ (x), at what values of x are the points of inflection?
Given f(x), for what intervals is
f “ (x) > 0?
Given f ‘ (x), for what intervals if f “ (x) positive?
Given f(x), when do f “ (x) and
f ‘ (x) have opposite signs?
-1.5, -0.5, 0.5, 1.5
-1, 0, 1
(-1.5, -0.5), (0.5, 1.5)
(-1, 0), (1, 2)
The graph is on the interval [-2, 2]
(-1.5, -1), (-0.5, 0)
(0.5, 1), (1.5, 2)

6. Given f(x), where is f ‘ (x) undefined?
Given f(x), write using interval notation the interval(s) on which the graph is concave down.
Given f ‘ (x), write using interval notation the interval(s) on which the graph is concave up.
Given f ‘’ (x), write using interval notation the interval(s) on which the graph is concave down.
(-1, 1)
(-1, 0), (1, 3)
None
-1, 1
The graph is defined on (-3, 3)

7. Given f(x), where is f ‘ (x) = 0?
Given f ‘ (x) write using interval notation the interval(s) on which the graph is increasing.
Given f ‘ (x), write using interval notation the interval(s) on which
f “ (x) is negative.
Given f “ (x) for what value(s) of x would there be a point of inflection?
-0.5, 0.5
(-1, 0), (1, 2]
(-0.5, 0.5)
–1, 0 1
The graph is defined on [-2, 2]

8. Given f ‘ (x), where is f ‘ (x) = 0?
Given f(x), write using interval notation the interval(s) on which f ‘ ‘ (x) is positive.
Given f ‘ (x), write using interval notation the interval(s) on which f ‘ ‘ (x) is negative.
Given f(x), at approximately what value of x would
f ‘ (x) = 1?
1, 3
(-2, 5)
(-2, 2)
2.5
The graph is defined on [-2, 5]

9. Given f(x), where is f ‘ (x) undefined?
Given f ‘ (x), where is f “ (x) undefined?
Given f “ (x), write using interval notation the interval(s) on which the graph would be concave up.
Given f “ (x), for what values of x is there a point of inflection?
(-1, 3] -1
The graph is defined on [-10, 3]

10. 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

11. 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