Section 2.6
Inflection Points and the Second Derivative
Calculator Required on all Sample Problems
f “ (a) gives us information on the concavity of the graph.
If f "  a   0 , the graph is concave up
Graph of f  x 
Graph of f '  x 
Slopes are positive
Graph of f "  x 
Coordinates are positive
f “ (a) gives us information on the concavity of the graph.
If f "  a   0 , the graph is concave down
Graph of f  x 
Graph of f '  x 
Slopes are negative
Graph of f "  x 
Coordinates are negative
If f ” (a) = 0, a point of inflection MAY exist
A point of inflection occurs when f “ changes signs.
Pt. of Inflection
NO Pt. of Inflection
A. Find the x-coordinate(s) of
the point(s) of inflection.
2
B. On what interval(s) is
f “ < 0?
(-1, 2)
C. On what interval(s) is
f ‘ > 0?
(-1, 1), (3, 5)
D. Where is a relative minimum
of f?
This is the graph of f(x) on (-1, 5)
x=3
A. Find the x-coordinates of the
point(s) of inflection
1, 3
B. On what interval(s) is
f “ < 0?
(1, 3)
C. On what interval(s) is
f ‘ > 0?
(0, 5)
D. Where is a relative minimum
of f?
This is the graph of f '  x  on (-1, 5)
x=0
A. Find the x-coordinates of the
point(s) of inflection
0
B. On what interval(s) is
f “ < 0?
(-1, 0)
C. On what interval(s) is
f ‘‘ > 0?
(0, 5)
This is the graph of f "  x  on (-1, 5)
A. Where is/are the point(s) of
inflection of f?
x = -1.5, x = -0.5, x = 0.5, x = 1.5
B. On what interval(s) is
f ‘ increasing?
(-1.5, -0.5), (0.5, 1.5)
C. On what interval(s) is
f “ < 0?
(-2, -1.5), (-0.5, 0.5), (1.5, 2)
D. On what interval(s) do
This is the graph of f(x) on [-2, 2].
f “and f ‘ have opposite
signs?
(-1.5, -1), (-0.5, 0)
(0.5, 1), (1.5, 2)
A. Where is/are the point(s) of
inflection of f?
x = -1, x = 0, x = 1
B. On what interval(s) is
f increasing?
(-2, -1.5), (-0.5, 0.5), (1.5, 2)
C. On what interval(s) is
f “ < 0?
(-2, -1), (0, 1)
D. On what interval(s) do
This is the graph of f '  x  on [-2, 2].
f “ and f ‘ have opposite
signs?
(-2, -1.5), (-1, -0.5)
(0, 0.5), (1, 1.5)
A. Where is/are the point(s) of
inflection of f?
x = -1.5, x = -0.5, x = 0.5, x = 1.5
B. On what interval(s) is
f concave up?
(-2, -1.5), (-0.5, 0.5), (1.5, 2)
C. On what interval(s) is
This is the graph of f "  x  on [-2, 2].
f “ < 0?
(-1.5, -0.5), (0.5, 1.5)
A. For what value(s) of x is f ‘
undefined?
-1, 1
B. On what interval(s) is f
concave down?.
(-1, 1)
C. On what interval(s) is f ‘
increasing?
(-3, -1), (1, 3)
This is the graph of f(x) on (-3, 3)
D. On what interval(s) is
f '  0?
(-3, -1), (0, 1)
A. For what value(s) of x is f '
undefined?
none
B. On what interval(s) is f
concave down?.
(-3, -1), (0, 1)
C. On what interval(s) is f '
increasing?
(-1, 0), (1, 3)
This is the graph of f '  x  on (-3, 3)
D. On what interval(s) is
f '  0?
none
A. On what interval(s) is f
concave up?
(-3, -1), (-1, 1), (1, 3)
B. List the value(s) of x for
which f has a point of
inflection.
none
C. For what value(s) of x is
f "x  0 ?
This is the graph of f "  x  on (-3, 3)
-1, 1
A. For what value(s) of x is
f ‘ (x) = 0?
-0.5, 0.5
B. On what interval(s) is
f ‘ > 0?
(-2, -0.5), (0.5, 2)
C. On what interval(s) is
f “ < 0?
(-2, 0)
D. Find the x-coordinate of the
This is the graph of f(x) on (-2, 2)
point(s) of inflection.
0
A. For what value(s) of x is
f ‘ (x) = 0?
-1, 0, 1
B. On what interval(s) is
f decreasing?
(-2, -1), (0, 1)
C. On what interval(s) is
f “ < 0?
(-0.5, 0.5)
This is the graph of f ‘ (x) on
(-2, 2).
D. Find the x-coordinate of the
point(s) of inflection of f.
-0.5, 0.5
A. On what interval(s) is f(x)
concave up?
[-1, 1), (3, 5]
B. Find the x-coordinate of the
point(s) of inflection of f.
1, 3
C. On what interval(s) is
f “ > 0?
[-1, 1), (3, 5]
This is the graph of f “ (x) on
[-1, 5].
For what value(s) of x does
f ‘ not exist?
none
On what interval(s) is f
concave down?
none
On what interval(s) is
f “ > 0?
(-10, 0), (0, 3)
This is the graph of f ‘ (x)
on [-10, 3].
Where is/are the relative
minima of f on [-10, 3]?
x = -10, x = -1
CALCULATOR REQUIRED
Which of the following is/are true about the function f if its
2
derivative is defined by f '  x    x  1  4  x  ?
increasing
I) f is decreasing for all x < 4
NO
II) f has a local maximum at x = 1
TRUE
III) 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
CALCULATOR REQUIRED
The graph of the second derivative of a function f is shown
below. Which of the following are true about the original function
f?
NO
I) The graph of f has an inflection point at x = -2
II) The graph of f has an inflection point at x = 3
YES
III) The graph of f is concave down on the interval (0, 4) NO
A) I only B) II only C) III only D) I and II only E) I, II and III
CALCULATOR REQUIRED
Which of the following statements are true about the function
3
f, if it’s derivative f ‘ is defined by 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 = 0 Use a = 2
III) 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 only
E) I, II and III
I) f '  4   4  4  2   0
YES
II) f '  0   0  0  2   0
YES
3
3
III) f '  x   x  x  2   Graph f "  x 
3
NO