Graphics Reference
In-Depth Information
Property (i)
fails
a
b
a
b
Property (ii)
fails
a
b
a
b
Property (iii)
fails
a
b
a
b
Figure9.2
If an
n
-times differentiable function has
n
1 zeros, then between the
extreme roots there is a point
c
at which
f
(
c
)
0.
11
(i) Continuous and differentiable by qns 8.15 and 8.17 since the
denominator is never 0.
f
(0)
. Rolle's Theorem fails.
Completeness is needed for the Maximum
—
minimum theorem.
(ii) Polynomials continuous and differentiable.
g
(1)
g
(4)
0.
g
is
bounded below but does not attain its lower bound. Domain not
an interval, i.e. not complete.
f
(2)
12
(i) By qn 6.23. (ii) By qns 7.31, 7.32 and 7.34. (iii) By qns 8.10 and
8.11.
(iv) By qn 8.6 since two derivatives are unequal.
Apply Rolle's Theorem to
g
on [
a
,
b
] then, for some
c
in (
a
,
b
),
g
0. Thefunction
f
in qn 8.22 is differentiable at every point.
For this function,
f
(
c
)
(
x
)
2
x
· sin(1/
x
)
cos(1/
x
), when
x
0, and
