Equation of line isy f(a) = {f(b) f(a)}/(b-a) . This means that we can apply the Mean Value Theorem for these two values of \(x\). The mean value theorem is the special case of Cauchy’s mean value theorem when
g
(
t
)
=
t
{\displaystyle g(t)=t}
.
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Define
There exists
c
(
a
,
b
)
{\displaystyle c\in (a,b)}
such that
D
(
c
)
=
0
{\displaystyle D'(c)=0}
.
Cauchy’s mean value theorem, also known as the extended mean value theorem,6 is a generalization of the mean value theorem.
For example, consider the following 2-dimensional function defined on an
n
{\displaystyle n}
-dimensional cube:
Then, by symmetry it is easy to see that the mean value of
G
{\displaystyle G}
over its domain is (0,0):
However, there is no point in which
G
=
(
0
,
0
)
{\displaystyle G=(0,0)}
, because
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|
Going Here
G
|
=
my review here 1
{\displaystyle |G|=1}
everywhere. .