mandel (tgm245) – HW10 – Radin – (56470)
1
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001
10.0 points
Let
f
be the function defined by
f
(
x
) = 3 + 2
x
1
/
3
.
Consider the following properties:
002
10.0 points
If
f
is increasing and its graph is concave
up on (0
,
1), which of the following could be
the graph of the
derivative
,
f
′
, of
f
?
A. derivative exists for all
x
;
B. has vertical tangent at
x
= 0 ;
C. concave up on (0
,
∞
)
.
Consequently,
A. not have: (
f
′
(
x
) = (2
/
3)
x
−
2
/
3
, x
negationslash
= 0);
B. has: (see graph);
C. not have: (
f
′′
(
x
)
<
0
,
x >
0).

Which does
f
have?

mandel (tgm245) – HW10 – Radin – (56470)
2
4.
f
′
(
x
)
1
Explanation:
The function
f
increases when
f
′
>
0 on
(0
,
1), and its graph is concave up when
f
′′
>
0. Thus on (0
,
1) the graph of
f
′
lies above the
x
-axis and is increasing. Of the four graphs,
only
1
f
′
(
x
)
has these properties.
003
10.0 points
When Sue uses first and second derivatives
to analyze a particular continuous function
y
=
f
(
x
) she obtains the chart
y
y
′
y
′′
x <
−
3
+
−
x
=
−
3
4
0
Consequently,
A.
f
is concave up on (
−∞
,
0)
.
B.
f
has a point of inflection at
x
= 0.
C.
f
is concave up on (0
,
2)
.
1.
all of them
2.
C only
3.
A and B only
4.
B and C only
correct
5.
A and C only
6.
B only
7.
none of them
8.
A only
Explanation:
The graph of
f
must look like
2
−
2
−
4
2
4
004
10.0 points

Which of the following can she conclude from
her chart?
The figure below shows the graphs of three
functions:

mandel (tgm245) – HW10 – Radin – (56470)
3
One is the graph of a function
f
, one is its
derivative
f
′
, and one is its second derivative
f
′′
.
Identify which graph goes with which
function.
1.
f
:
f
′
:
f
′′
:
correct
2.
f
:
f
′
:
f
′′
:
3.
f
:
f
′
:
f
′′
:
4.
f
:
f
′
:
f
′′
:
5.
f
:
f
′
:
f
′′
:
6.
f
:
f
′
:
f
′′
:
Explanation:
Calculus tells us that
f
(i) has horizontal tangent at (
x
0
, f
(
x
0
))
when
f
′
crosses the
x
-axis,
(ii) is increasing when
f
′
>
0, and
(iii) is decreasing when
f
′
<
0,
(iv) has a local max at
x
0
when
f
′
(
x
0
) = 0
and
f
′′
(
x
0
)
<
0,
(v) has a local min at
x
0
when
f
′
(
x
0
) = 0
and
f
′′
(
x
0
)
>
0,
(vi) is concave up when
f
′′
>
0,
(v) and concave down when
f
′′
<
0.
Inspection of the graphs thus shows that
f
:
f
′
:
f
′′
:
.
005
10.0 points
Find all intervals on which
f
(
x
) =
x
2
(
x
+ 2)
3
is decreasing.
1.
(
−∞
,
−
2)
,
(
−
2
,
0]
,
[4
,
∞
)
correct
2.
[
−
4
,
0]
3.
(
−∞
,
−
4]
,
[0
,
∞
)
4.
[0
,
4]
5.
(
−∞
,
0]
,
[4
,
∞
)
6.
(
−∞
,
−
4]
,
[0
,
2)
,
(2
,
∞
)
7.
(
−
2
,
2)
Explanation:
By the Quotient Rule,
f
′
(
x
) =
2
x
(
x
+ 2)
3
−
3
x
2
(
x
+ 2)
2
(
x
+ 2)
6
=
2
x
(
x
+ 2)
−
3
x
2
(
x
+ 2)
4
=
x
(4
−
x
)
(
x
+ 2)
4
.
Now
f
will be decreasing on an interval [
a, b
]
(resp. [
a, b
)) when
f
′
(
x
)
<
0 on the interval
(
a, b
) (resp. [
a, b
) if
f
(
b
) is not defined). In
this case the inequality
f
′
(
x
)
<
0 will hold
when
x
(4
−
x
)
<
0
,
x
negationslash
=
−
2
.