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Mathematics
Mathematics
Problem
Let
f
(
x
)
=
x
3
f(x)=x^{3}
f
(
x
)
=
x
3
on the interval
[
1
,
2
]
[1,2]
[
1
,
2
]
. a. (i) Show that there exists a number
ξ
\xi
ξ
in
(
1
,
2
)
(1,2)
(
1
,
2
)
such that
f
′
(
ξ
)
f'(\xi)
f
′
(
ξ
)
equals the average rate of change of
f
f
f
on
[
1
,
2
]
[1,2]
[
1
,
2
]
. b. (ii) Find one value of
ξ
\xi
ξ
that satisfies this condition.
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