1
Math 131 WIR, copyright Angie Allen
Math 131 Week-in-Review #8 (Sections 4.2, 4.3, and 4.6)
1. Use the graph below to state the absolute and local maximum and minimum values of the function f on the following
intervals, if any exist. (Graph courtesy of Kendra Kilmer)
a) (−∞, ∞)
4
3
b) [−0.5, 1)
2
f(x)
1
−4
−3
−2
−1
1
−1
c) [3, 4.5)
−2
−3
−4
2. Find the critical numbers of each of the functions below.
a) f (x) = |5x − 8|
b) g(x) =
p
5 2
x + x − 12
2
3
4
Math 131 WIR, copyright Angie Allen
2
3. Use calculus to determine where the function f (x) = x1/3 − x−2/3 is increasing/decreasing, as well as the values of any
local extrema. Remember to classify the type of local extrema, if necessary.
−x − 2
2x + 10
and f ′′ (x) =
.
(x − 4)3
(x − 4)4
Find where any local maximum and minimum values of f (x) occur using the Second Derivative Test, if possible. If it
is not possible, then use the First Derivative Test.
4. Consider a function f (x) that is continuous on its domain of (−∞, 4)∪(4, ∞). Also, f ′ (x) =
3
Math 131 WIR, copyright Angie Allen
5. Use calculus, if possible, to find the absolute maximum and minimum values of the function f (x) =
x−2
on each
(x − 4)2
of the following intervals, if they exist.
a) [−2, 3]
b) [−5, 5]
6. Consider a function f (x) that is continuous everywhere. Also, f ′ (x) =
where f (x) is concave upward/downward and has any inflection points.
2x
(x2 + 3)2
and f ′′ (x) =
6 − 6x2
. Determine
(x2 + 3)3
Math 131 WIR, copyright Angie Allen
4
7. A rectangular storage container with an open top is to have a volume of 10 m3 . The length of its base is twice the width.
Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Use calculus to find
the cost of materials for the cheapest such container. (Source: #14, pg. 306, Stewart)
8. If f is continuous on (−∞, ∞), and we know f ′ (3) = 0, f ′′ (3) = 7, and f ′′ (x) is continuous near x = 3, what can we
conclude about f ? What if f ′′ (3) = 0?
5
Math 131 WIR, copyright Angie Allen
2
9. Use calculus to find any asymptotes of the function f (x) = xe−8x , where the function is increasing/decreasing and
concave upward/downward, the values of any local extrema, and any inflection points. Use your information to graph
f (x).
Math 131 WIR, copyright Angie Allen
6
10. A department store wants to create rectangular flyers to advertise its upcoming holiday sale. The flyer will have 60
square inches of print, and the margin on each side needs to be 1.5 inches. Use calculus to find the dimensions of the
flyer so that the least amount of paper is used.
11. Briefly state the difference between partition numbers and critical numbers of a function f .
7
Math 131 WIR, copyright Angie Allen
12. Use the graph of f (x) below to find the following:
a) Where is f (x) < 0?
y
b) Find the critical numbers of f (x).
a
c) Where is f ′′ (x) > 0?
d) Find the partition numbers of f ′ (x).
e) Where is f ′ (x) < 0?
f) Where is f ′ (x) decreasing?
g) Find and classify any local extrema of f (x).
b
c d e
x
Math 131 WIR, copyright Angie Allen
8
13. Find the critical numbers of the functions below.
a) x3 e−5x
b) x4/5 (x − 9)2
14. Consider a function f that is smooth and continuous on its domain of (−∞, 2) ∪ (2, ∞).
8(x + 4)
. Find any critical numbers of f (x), the intervals where f (x) is increasing/decreasing, as well
Also, f ′ (x) =
(x − 2)4
as where any local extrema occur (be sure to specify the type of any local extrema).
Math 131 WIR, copyright Angie Allen
9
15. Find the absolute maximum and minimum values, if any exist, of the function f (x) = x − ln(3x) on the interval [0.5, 2].
16. List the three conditions necessary for a function f (x) to have a local maximum or minimum at x = c.
1.
2.
3.
10
Math 131 WIR, copyright Angie Allen
17. Determine where the function f (x) =
−1
x2 + 3
is concave upward/downward and has any inflection points.
18. A box with a square base and open top must have a volume of 32,000 cm3 . Find the dimensions of the box that
minimize the amount of material used.
11
Math 131 WIR, copyright Angie Allen
19. Assuming the graph shown below is f ′ (x) and f (x) is continuous on its domain of (−∞, d) ∪ (d, ∞), find the following:
a) Find the critical numbers of f (x).
y
x
b) Find the partition numbers of f ′ (x).
a
b
c
d
e
c) Where is f (x) increasing?
d) Find and classify any local extrema of f (x).
e) Where is f (x) concave up?
f) Where is f ′′ (x) < 0?
g) Find the partition numbers of f ′′ (x).
h) Where does f (x) have inflection points (if any)?
i) Find the critical numbers of f ′ (x).
f g
Math 131 WIR, copyright Angie Allen
12
20. Use calculus to find where the function f (x) = x5 ln x is increasing/decreasing and concave upward/downward, the
values of any local extrema, and any inflection points.
Math 131 WIR, copyright Angie Allen
13
21. For each of the following functions, find any critical numbers, the intervals where the function is increasing/decreasing,
and any local extrema. Be sure to classify the type of each extrema.
a) f (x) = x − ln x
b) f (x) = x2 e−x
14
Math 131 WIR, copyright Angie Allen
22. Assuming the graph shown is f ′′ (x), and f (x) is continuous on its domain of (a, c) ∪ (c, ∞) and f ′ (x) is continuous on
its domain of (a, c) ∪ (c, ∞), find the following:
a) Find the partition numbers of f ′′ (x).
y
b) Where is f ′ (x) increasing?
x
a
c) Where is f (x) concave down?
d) Where does f (x) have inflection points (if any)?
e) Find f
′′′ (e).
f) Find and classify any local extrema of f ′ (x).
b
c
d
e f