Lescure, Etienne – Review 2 – Due: Dec 10 2007, 10:00 pm – Inst: Diane Radin
2
Consider the following properties:
A. has local maximum at
x
= 0
B. concave down on (
∞
,
0)
∪
(0
,
∞
)
Which does
f
have?
1.
both of them
2.
neither of them
3.
A only
correct
4.
B only
Explanation:
The graph of
f
is
2
4

2

4
2
4
On the other hand, after differentiation,
f
0
(
x
) =

2
3
x
1
/
3
,
f
00
(
x
) =
2
9
x
4
/
3
.
Consequently,
A. TRUE: see graph
B. FALSE:
f
00
(
x
)
>
0
,
x
6
= 0
.
keywords:
concavity,
local
maximum,
True/False, graph
004
(part 1 of 1) 10 points
The derivative,
f
0
, of
f
has graph
a
b
c
graph of
f
0
Use it to locate the critical point(s)
x
0
at
which
f
has a local minimum?
1.
x
0
=
a , b , c
2.
x
0
=
c
3.
x
0
=
b
4.
x
0
=
b , c
5.
x
0
=
a
correct
6.
x
0
=
c , a
7.
x
0
=
a , b
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 first 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
.
Consequently, by looking at the sign of
f
0
(
x
)
near each of
x
0
=
a, b,
and
c
we see that
f
has
a local minimum only at
x
0
=
a
.