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Calculus
3.1-3.4 Review
Name:____________________________________
Mitchell
o Show your work whenever possible and circle your final answers.
1) True or False:
If f’’(c)=0, then (c,f(c)) is a point of inflection.
Justify your answer.
2) True or False:
If f(c) is a local maximum of a continuous
function f on an open interval (a,b), then
f ’(c)=0.
Justify your answer.
3) True or False:
If f’(c)=0 and f’’(c)<0, then f(c) is a local
maximum.
Justify your answer.
4) Find the critical values of f (x) = x − x − 2 .
x2
5) Find the critical values of f (x) =
.
5x + 4
6) How many critical values does the function
7) Which of the following values is the absolute
3
5
4
f (x) = (x − 2) (x + 3) have?
(A)
(B)
(C)
(D)
(E)
One
Two
Three
Five
Nine
2
maximum of the function f (x) = 4x − x 2 + 6 on
the interval [0,4]?
(A)
(B)
(C)
(D)
(E)
0
2
4
6
10
8) Find the extrema on [-1,3] for f where
f (x) = 31 x 3 + 2x 2 − 5x + 1 .
x+1
.
x−1
Find all values of c that satisfy the conclusion
of the mean value theorem on [ 23 ,5].
10) Let f be the function given by f (x) =
3
2
9) Let f be the function given by
f (x) = x 3 − 3x 2 + 2x . Find all values of c that
satisfy the conclusion of the mean value
theorem on [−1, 1].
11) If f '(x) = x(x − 3)4 (x + 5)7 , at which values of x
is f(x) a relative minimum value?
x4
3
5
12) x = t3 − 3t 2 and y = t2 − 2t . Find the x and ycoordinates for each critical value on the curve
and identify each point as having a vertical or
horizontal tangent line.
13) If f (x) =
at x=
14) Given the function defined by
f (x) = 3x 5 − 20x 3 , find all values of x for which
the graph of f is concave up.
15) The function f given by f (x) = 2x 3 − 3x 2 − 12x
has a relative minimum at x=
A.
B.
C.
D.
E.
x>0
− 2 < x < 0 or x > 2
−2 < x < 0 or x > 2
x> 2
−2 < x < 2
− x5 , f ‘(x) attains a maximum value
A. -1
B. 0
C. 1
D. 43
E. 53
A. -1
B. 0
C. 2
3 − 105
D.
4
3 + 105
E.
4
16) At what value of x does the graph of
1
1
y = 2 − 3 have a point of inflection?
x
x
A.
B.
C.
D.
E.
17) Find the x-coordinates of all inflection points of
x 8 9x 6
f if f (x) =
.
−
56 30
0
1
2
3
At no value of x
18) Given the function f defined by
f (x) = cos x − cos 2 x for −π ≤ x ≤ π .
19) If y = 2x 3 + 4ax 2 + bx + 3 has an inflection
point at (2,-3), find a and b.
a) Find the x-intercepts of the graph of f.
b) Find the x and y-coordinates of all relative
maximum and minimum points of f.
(justify your answers)
c) Find the intervals on which the graph of f is
increasing.
20) The graph of f, the derivative of f is shown below. On which of the following intervals is f decreasing?
A.
B.
C.
D.
E.
[2,4] only
[3,5] only
[0,1] and [3,5]
[2,4] and [6,7]
[0,2] and [4,6]
y
f’(x)
x
7
21) The graph of f ’(x) is shown to the right. For what
value of x does f(x) have a relative maximum?
a)
b)
c)
d)
e)
1
2
3
4
5
y=f ‘(x)
0
1
2
3
4
(2,4) only
(1,3) and (5, ∞ )
(- ∞ ,1) and (3,5)
(0,2) and (4,6)
(- ∞ ,0), (2,4) and (6, ∞ )
0
1
2
4
6
-1 7
y=f ‘‘(x)
-1 7
24) The graph of f ’(x) is shown to the right.
f has an inflection point at which of the
following x-values?
a)
b)
c)
d)
e)
y=f ‘(x)
23) The graph of f ’’(x) is shown to the right.
Use the graph of f ’’(x) to approximate
the intervals on which f(x) is concave up.
a)
b)
c)
d)
e)
7
22) The graph of f ’(x) is shown to the right.
For what value of x is f(x) concave down?
a)
b)
c)
d)
e)
-1 y=f ‘(x)
-1 7
25) The function f and its derivatives have the properties indicated in the table below.
x
f(x)
f’(x)
f’’(x)
− 2 < x < −1
+
+
-
-1
2
0
0
0
1
UND
UND
−1 < x < 0
+
-
1
0
0
0
0<x<1
+
+
1<x<2
-
2
-1
UND
UND
2<x<3
+
-
a) Find the x-coordinates of each point at which f attains an absolute maximum value or an absolute
minimum value. For each x-coordinate you give, state whether f attains an absolute minimum or
an absolute maximum.
b) Find the x-coordinate of each point of inflection on the graph of f. Justify your answer.
c) Sketch a graph of a function with all of the given characteristics.
26) The figure below shows the graph of f’, the derivative of a function f. The domain of the function f is
the set of all x such that −3 ≤ x ≤ 3 .
a) For what values of x, −3 ≤ x ≤ 3 , does f have a relative maximum? A relative minimum? Justify
your answers.
b) For what values of x is the graph of f concave up? Justify your answer.
c) Use the information found in parts a and b and the fact that f ( −3) = 0 to sketch a possible graph
of f.
y = f’(x)
-3 7