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Villarreal, Natalie – Homework 10 – Due: Oct 30 2007, 3:00 am – Inst: Louiza Fouli
1
This printout should have 21 questions.
Multiplechoice questions may continue on
the next column or page – fnd all choices
be±ore answering.
The due time is Central
time.
001
(part 1 o± 1) 10 points
The derivative o± a ±unction
f
is given by
f
0
(
x
) = (
x
2
+ 3
x

10)
g
(
x
)
±or some unspecifed ±unction
g
such that
g
(
x
)
>
0 ±or all
x
. At which point(s) does
f
have a local minimum?
1.
local minimum at
x
=

5
2.
local minimum at
x
=

2
3.
local minimum at
x
= 5
4.
local minimum at
x
=

5
,
2
5.
local minimum at
x
= 2
correct
Explanation:
At a local minimum o±
f
, the derivative
f
0
(
x
) will be zero,
i.e.
,
(
x

2)(
x
+ 5)
g
(
x
) = 0
.
Thus the critical points o±
f
occur only at
x
=

5
,
2. To classi±y these critical points
we use the First Derivative test; this means
looking at the sign o±
f
0
(
x
). But we know that
g
(
x
)
>
0 ±or all
x
, so we have only to look at
the sign o± the product
(
x

2)(
x
+ 5)
o± the other two ±actors in
f
0
(
x
). Now the sign
chart

5
2
+
+

±or (
x

2)(
x
+ 5) shows that the graph o±
f
is increasing on (
∞
,

5), decreasing on
(

5
,
2), and increasing on (2
,
∞
).
Conse
quently,
f
has
a local minimum at
x
= 2
,
.
keywords: local minimum, First Derivative
Test critical points, sign chart, conceptual,
002
(part 1 o± 1) 10 points
Let
f
be the ±unction defned by
f
(
x
) = 5

x
2
/
3
.
Consider the ±ollowing properties:
A. derivative exists ±or all
x
;
B. concave down on (
∞
,
0)
∪
(0
,
∞
);
C. has local minimum at
x
= 0;
Which does
f
have?
1.
C only
2.
A only
3.
B and C only
4.
A and C only
5.
B only
6.
A and B only
7.
All o± them
8.
None o± them
correct
Explanation:
The graph o±
f
is
2
4

2

4
2
4
On the other hand, a±ter di²erentiation,
f
0
(
x
) =

2
3
x
1
/
3
,
f
00
(
x
) =
2
9
x
4
/
3
.
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View Full DocumentVillarreal, Natalie – Homework 10 – Due: Oct 30 2007, 3:00 am – Inst: Louiza Fouli
2
Consequently,
A. not have: (
f
0
(
x
) =

(2
/
3)
x

1
/
3
,
x
6
=
0;
B. not have: (
f
00
(
x
)
>
0
, x
6
= 0);
C. not have: (see graph).
keywords:
concavity,
local
maximum,
True/False, graph
003
(part 1 of 1) 10 points
Use the graph
a
b
c
of the derivative of
f
to locate the critical
points
x
0
at which
f
does not have a local
minimum?
1.
x
0
=
c, a
2.
x
0
=
b
3.
x
0
=
a, b
4.
x
0
=
a
5.
x
0
=
c
6.
x
0
=
a, b, c
7.
x
0
=
b, c
correct
8.
none of
a, b, c
Explanation:
Since the graph of
f
0
(
x
) has no ‘holes’,
the only critical points of
f
occur at the
x

intercepts of the graph of
f
0
,
i.e.
, at
x
0
=
a, b,
and
c
. Now by the ±rst derivative test,
f
will
have
(i) a local maximum at
x
0
if
f
0
(
x
) changes
from
positive
to
negative
as
x
passes
through
x
0
;
(ii) a local minimum at
x
0
if
f
0
(
x
) changes
from
negative
to
positive
as
x
passes
through
x
0
;
(iii) neither a local maximum nor a local
minimum at
x
0
if
f
0
(
x
) does not change
sign as
x
passes through
x
0
.
Consequently, by looking at the sign of
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