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Mathematics
Problem
Let
f
(
x
)
=
e
2
x
+
x
f(x)=e^{2x}+x
f
(
x
)
=
e
2
x
+
x
on the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
. Which value of
c
c
c
in
(
0
,
1
)
(0,1)
(
0
,
1
)
makes the instantaneous rate of change of
f
f
f
equal its average rate of change on the interval? a)
ln
(
e
2
−
1
2
)
\ln\!\left(\dfrac{e^{2}-1}{2}\right)
ln
(
2
e
2
−
1
)
b)
1
2
ln
(
e
2
+
1
2
)
\dfrac{1}{2}\,\ln\!\left(\dfrac{e^{2}+1}{2}\right)
2
1
ln
(
2
e
2
+
1
)
c)
1
2
ln
(
e
2
−
1
2
)
\dfrac{1}{2}\,\ln\!\left(\dfrac{e^{2}-1}{2}\right)
2
1
ln
(
2
e
2
−
1
)
d)
1
2
ln
(
e
2
2
)
\dfrac{1}{2}\,\ln\!\left(\dfrac{e^{2}}{2}\right)
2
1
ln
(
2
e
2
)
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