Graphics Reference
In-Depth Information
Answers
1 Yes. Either a local maximum or a local minimum for the function.
2 A differentiable function is continuous and a continuous function on a
closed interval is bounded and attains its bounds, from qns 7.31 and
7.34.
If
f
(
a
)
f
(
b
) is not a maximum then the maximum occurs on the open
interval. If
f
(
a
)
f
(
b
) is not a minimum then the minimum occurs on
the open interval. If
f
(
a
)
f
(
b
) is both a maximum and a minimum,
then the function is constant. Yes.
3 Yes, by qn 8.30.
4 If
f
(
a
)
f
(
b
)
M
m
, there is a
c
with
f
(
c
)
m
.
f
(
b
)
m
M
, then there is a
c
with
f
(
c
)
M
.
If
f
(
a
)
f
(
b
)
M
m
, then
f
(
x
)
M
m
for all
x
, and so
f
If
f
(
a
)
(
x
)
0
from qn 8.6.
5 See figure 9.1.
6 Differentiable
Continuous.
At a local maximum or minimum
f
(
x
)
0.
7
(i)
f
is bounded and attains its bounds.
(iii)
when
f
(
a
) is not a maximum, there is a maximum
f
(
c
) with
a
c
b
; when
f
(
a
) is not a minimum there is a minimum
f
(
c
)
with
a
b
; and when
f
(
a
) is a maximum and a minimum
f
(
c
)
is a maximum and a minimum.
c
(ii)
f
(
c
)
0.
8 See figure 9.2.
9 Apply Rolle's Theorem to the function
x
a
x
a
x
/2
a
x
/3
...
a
x
/(
n
1) on theintrval [0, 1].
10 Apply Rolle's Theorem to [
a
,
b
] and [
b
,
c
], to find
f
0 with
a
x
b
and
f
(
y
)
0 with
b
y
c
. Then apply Rolle's Theorem to
thefunction
f
on theintrval [
x
,
y
].
(
x
)
a
b
Figure9.1