Graphics Reference
In-Depth Information
21 For a continuous function
f
:[
a
,
b
]
[
a
,
b
] the existence of a root
for
f
(
x
)
x
is guaranteed, as we found in qn 7.20. Illustrate how, in
theabsenceof thecondition
1,
either
(a) multipleroots for
f
(
x
)
x
may exist,
or
(b) the sequence (
a
f
(
x
)
) may not converge.
22 A function
f
:[
a
,
b
]
R is continuous on the closed interval [
a
,
b
]
and differentiable on the open interval (
a
,
b
).
If, for some
c
, with
a
c
b
,
f
(
c
)
0, and
f
(
c
) exists and
0,
prove that there is a neighbourhood of
c
within which
f
(
x
)
0
when
x
c
, and
f
(
x
)
0 when
x
c
. Deduce that
f
has a local
minimum at
c
.
The result of qn 22 may seem deceptively straightforward. However,
thetwo conditions,
f
(
c
)
0 and
f
(
c
)
0, do not, by themselves, imply
the continuity of derivatives other than at
x
c
.
(23) Let real functions
f
: R
R be defined by
x
2!
x
3!
x
n
!
.
f
(
x
)
1
x
...
(
1)
Check that
f
f
.
If
n
is even and
f
is monotonic
decreasing and has one and only one root. The existence of at least
one root comes from the Intermediate Value Theorem as in qn
7.19.
Deduce that there is exactly one value of
x
at which
f
(
x
)
0 for all
x
, provethat
f
(
x
)
0,
that
f
is minimal at this point, and that
f
(
x
)
0atthis point
so that
f
is positive everywhere.
Now useinduction to provethat
f
is positivewhen
n
is even and
has a uniqueroot when
n
is odd.
24
(i) If thedomain of a real function
f
contains at least the closed
interval [
x
,
y
] and
x
y
, show that thepoint
(
x
f
(
x
)
)
f
(
y
)) lies on the line segment
joining thetwo points (
x
,
f
(
x
)) and (
y
,
f
(
y
)), provided
0
(1
)
y
,
(1
1.
Draw a diagram.
How would you describe the inequality
f
(
x
(1
)
y
)
f
(
x
)
(1
)
f
(
y
) according to your
diagram?
When such an inequality holds throughout the domain of
definition of a function and for all
, with 0
1, the
function is said to be
concave upwards
(or
convex downwards
).
