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Mathematics
Mathematics
Problem
Consider the function
f
(
x
)
=
e
2
x
f(x)=e^{2x}
f
(
x
)
=
e
2
x
on the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
. (a) Prove that there is exactly one
c
∈
(
0
,
1
)
c\in(0,1)
c
∈
(
0
,
1
)
for which
f
′
(
c
)
f'(c)
f
′
(
c
)
equals the average rate of change of
f
f
f
on
[
0
,
1
]
[0,1]
[
0
,
1
]
. (b) Calculate this value of
c
c
c
correct to four decimal places.
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