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
.