Graphics Reference
In-Depth Information
The Second and Third Mean Value Theorems
31
Second Mean Value Theorem
A function
f
:[
a
,
b
]
R has a continuous derivative on the closed
interval [
a
,
b
] and is twice differentiable on the open interval (
a
,
b
).
Let
f
(
b
)
f
(
a
)
(
b
a
)
f
(
a
)
(
b
a
)
K
.
Apply Rolle's Theorem to the function
F
:[
a
,
b
]
R
given by
F
(
x
)
f
(
b
)
f
(
x
)
(
b
x
)
f
(
x
)
K
(
b
x
)
to show that, for some
c
with
a
c
b
,
K
f
(
c
). Deduce that
(
b
a
)
2
f
(
b
)
f
(
a
)
(
b
a
)
f
(
a
)
f
(
c
).
32
Rewrite the Second Mean Value Theorem
replacing
b
by
a
h
.
Then, by a judicious choice of
a
and
h
, show that for any twice
differentiable function
f
:
R
R
x
2
f
(
c
) for some
c
(0,
x
).
f
(
x
)
f
(0)
xf
(0)
33
Third Mean Value Theorem
If a function
f
:[
a
,
b
]
has a continuous second derivative on the
closed interval [
a
,
b
] and is three times differentiable on the open
interval (
a
,
b
), then there exists a
c
, with
a
c
b
, such that
R
(
b
a
)
2!
(
b
a
)
3!
f
(
b
)
f
(
a
)
(
b
a
)
f
(
a
)
f
(
a
)
f
-
(
c
).
Indicate how to prove the Third Mean Value Theorem by defining
a constant
K
and a function
F
analogous to thosein qn 31, and
applying Rolle's theorem.
Reformulate the theorem, replacing
b
by
a
h
.
34 If
f
(
x
)
a
a
x
a
x
...
a
x
, provethat
f
(0)
n
!
a
f
(0),
a
f
(0),
a
f
(0), . . .,
a
.
We now extend the Second and Third Mean Value Theorems to an
n
th Mean Value Theorem to compare functions, which may be
differentiated any number of times, with polynomials.
