white (taw933) – Review 2 – benzvi – (55600)
1
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printout
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have
18
questions.
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001
10.0 points
Which one of the following properties does
f
(
x
) =
x

2
x
2
+ 12
have?
1.
local min at
x
=

6
2.
local max at
x
= 6
correct
3.
local max at
x
= 2
4.
local max at
x
=

6
5.
local min at
x
= 2
6.
local min at
x
= 6
Explanation:
By the Quotient rule,
f
(
x
) =
x
2
+ 12

2
x
(
x

2)
(
x
2
+ 12)
2
=
12 + 4
x

x
2
(
x
2
+ 12)
2
.
The critical points of
f
occur when
f
(
x
) = 0,
i.e.
, at the solutions of
f
(
x
) =
(2 +
x
)(6

x
)
(
x
2
+ 12)
2
= 0
.
Thus the critical points of
f
are
x
=

2 and
x
= 6.
To classify these critical points we use the
First Derivative Test. But the sign of
f
de
pends only on the numerator, so it is enough,
therefore, to look only at a sign chart for
(2 +
x
)(6

x
):

2
6


+
From this it follows that
f
is decreasing on
(
∞
,

2),
increasing on (

2
,
6),
and de
creasing on (6
,
∞
). Consequently,
f
has a
local maximum at
x
= 6
.
keywords:
local maximum, local minimum,
critical point, quotient rule, First Derivative
Test, rational function
002
10.0 points
Determine if the limit
lim
x
→ ∞
x
+ 3
x
2

4
x
+ 5
exists, and if it does, find its value.
1.
limit =
3
5
2.
limit = 5
3.
limit = 0
correct
4.
limit = 3
5.
limit doesn’t exist
6.
limit =

1
4
Explanation:
Dividing in the numerator and denominator
by
x
2
, the highest power, we see that
x
+ 3
x
2

4
x
+ 5
=
1
x
+
3
x
2
1

4
x
+
5
x
2
.
On the other hand,
lim
x
→ ∞
1
x
=
lim
x
→ ∞
1
x
2
= 0
.
By Properties of limits, therefore, the limit
exists and
limit = 0
.