Practice on Critical Numbers & Extrema
Calculus Sections 3.1 – 3.3
1.
Find all critical numbers of

f ( x)  9  x
A.
B.
C.
D.
E.
.
3
2 5
0
3
-3, 3
-3, 0, 3
none of these
2. Determine from the graph whether f
possesses extrema on the interval (a, b).
4. Find the values of x that produce relative
extrema for the function
f(x) = (x + 1)2(x – 2).
A. Relative maximum at x = -1; Relative
minimum at x = 1.
B. Relative maximum at x = 1, x = 3;
Relative minimum at x = -1.
C. Relative minimum at x = 2.
D. Relative maximum at x = -1; Relative
minimum at x = 2
E. None of these
y
5. Use the graph to identify the open intervals
where the function is increasing or
decreasing.
x
a
A.
B.
C.
D.
E.
c
b
Maximum at x = c, minimum at x = b
Maximum at x = c, no minimum
No maximum, minimum at x = b
No extrema
None of these
3. Which of the following statements is true
of f(x) = -x3 – 6x2 – 9x – 2?
A. f is decreasing on (-3, -1).
B. f is increasing on (-3, -1).
C. f is increasing on (-∞, -3).
D. f is increasing on (-2, ∞)
E. None of these
y
x
A. Increasing (-∞, 1) and (-3, ∞);
decreasing (-3, 1)
B. Increasing (0, 2); decreasing (-∞, 0)
and (2, ∞)
C. Increasing (-∞, ∞)
D. Increasing (-∞, 0) and (2, ∞);
decreasing (0, 2)
E. None of these
8. What is the sign of the second derivative of
f at the indicated point?
8
6. If f ( x )   , which statement is correct?
x
A. The graph of f is concave up on the
interval (0, ∞).
B. The graph of f is concave up on the
interval (-∞, 0).
C. The graph of f is concave down on the
interval (-∞, 0)
D. The graph of f is concave up on the
interval (-∞, ∞).
E. None of these
y
x
A.
B.
C.
D.
E.
7. Find all points of inflection of the function
f(x) = x4 + x3.
1
 1
A. (0, 0) and   ,  
 2 16 
Positive
Negative
Zero
The sign cannot be determined
None of these
1
 1
B.   ,  
 2 16 
C. (0, 0)
27 
 3
D. (0, 0) and   , 

 4 256 
E. None of these
3
9. Use a graphing calculator to graph f ( x)  x 5 . Use the graph of f to determine the open intervals
where the graph of f is concave upward or concave downward.
10. Find the critical numbers of f(x) = x3 – 12x2.
11. Find the minimum and maximum values of f(x) = x2 – 2x + 1 on the interval [0, 3].
12. Find the open intervals on which f ( x) 
1
is increasing or decreasing.
x2
13. Find the relative minimum and relative maximum for f(x) = 2x3 + 3x2 – 12x.
14. Let f(x) = x4 – 4x3 + 4x2 + 1. Respond to each part below.
a. Find all critical numbers of f.
b. Find the intervals over which f is increasing or decreasing.
c. Locate relative extrema of f using the first derivative test.
d. Find the y-intercept of f.
e. Sketch the graph of the curve.
15. Find the intervals on which the graph of the function f(x) = x4 – 4x3 + 2 is concave upward or concave
downward. Then find all points of inflection for the function.
16. The graph of a polynomial function, f, is given. On the same coordinate axes, sketch f ′ and f ″.
y
x
Calculus Practice: 3.1-3.4
1. Find the absolute extrema in each interval.
a) f(x) = x3 – 12x
on [0, 4]
c)
e)
b) h(t ) 
t
on [3, 5]
t2
d)
2  x 2 , 1  x  3
on [1, 5]
f ( x)  
2  3x, 3  x  5
2. Use your understanding of each theorem to respond to each prompt.
a) Sketch a function which is continuous on [a, b] b) The Mean Value Theorem states: If f is
where Rolle’s Theorem does not apply.
continuous on [a, b] and differentiable on (a, b),
then there exists a number, c, in (a, b) such that…
f ′(c) =
a
b
3. Find the value of c guaranteed by Rolle’s Theorem. If no such value exists, explain why not.
6x
b) f ( x)  3  x  3 on [0, 6]
 
a) f ( x) 
 4 sin 2 x on 0, 

 6
4. Find the value of c guaranteed by the Mean Value Theorem. If no such value exists, state why not.
f ( x) 
x2
on [2, 6]
5. Find all critical numbers and intervals where the function is increasing.
x3
a) f(x) = x4 – 32x + 4
c) h(x) = x4 – 2x3
b) g ( x)  3
x
6. Find all inflection points and intervals where the function is concave upward.
a) f(x) = 5 + 3x2 – x3
b) g ( x)  x 2  1
c) h(x) = sin x + cos x
where 0 < x < 2π