Graphics Reference
In-Depth Information
R
3 If
f
:[
a
,
b
]
is differentiable and, for some
c
such that
a
c
b
,
f
(
x
)
f
(
c
) for all
x
[
a
,
b
], must
f
(
c
)
0? Why?
R
4 If
f
:[
a
,
b
]
is differentiable,
f
(
a
)
f
(
b
),
M
sup
f
(
x
)
a
x
b
,
and
m
inf
f
(
x
)
a
x
b
,
useqn 2 to show that threis a
c
with
a
b
for which
f
(
c
)
M
,ora
c
for which
f
(
c
)
m
, or possibly distinct
c
s satisfying
each of these conditions.
Deduce from qn 3 that
f
(
c
)
0.
c
5 Sketch the graph of a continuous function
f
:[
a
,
b
]
R
which is differentiable on the open interval (
a
,
b
), but not
differentiable at the points
a
or
b
.
6 Identify the two places in the proof of the result of qn 4 at which
the differentiability of the function
f
has been invoked.
7
Rolle
'
s Theorem
If thefunction
f
:[
a
,
b
]
R satisfies the conditions
(i)
f
is continuous on [
a
,
b
],
(ii)
f
is differentiable on (
a
,
b
),
(iii)
f
(
a
)
f
(
b
),
show that there is a point
c
in the open interval (
a
,
b
) such that
f
(
c
)
0.
8 Sketch the graphs of functions for which one or more of the
conditions (i), (ii) and (iii) in qn 7 fail, illustrating how the
conditions are necessary for Rolle's Theorem, and also how, even in
their absence, there may still exist a point
c
(
a
,
b
) for which
f
(
c
)
0.
9 If
a
a
/2
a
/3
...
a
/(
n
1)
0, provethat the
polynomial function
x
a
a
x
a
x
...
a
x
has a zero in the interval [0, 1].
10 If thefunction
f
: R
R is twice differentiable and
f
(
a
)
f
(
b
)
f
(
c
)
0, with
a
b
c
, provethat for some
d
(
a
,
c
),
f
(
d
)
0. Generalise.
