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Mathematics
Mathematics
Problem
For a positive constant
t
t
t
, define
F
t
(
x
)
=
e
t
x
−
x
−
2
F_t(x)=e^{tx}-x-2
F
t
(
x
)
=
e
t
x
−
x
−
2
on the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
. For which of the following conditions on
t
t
t
is it guaranteed that the equation
e
t
x
=
x
+
2
e^{tx}=x+2
e
t
x
=
x
+
2
has at least one solution with
x
x
x
in
(
0
,
1
)
(0,1)
(
0
,
1
)
? a)
0
<
t
<
ln
3
0<t<\ln 3
0
<
t
<
ln
3
b)
t
=
ln
3
t=\ln 3
t
=
ln
3
c)
t
>
ln
3
t>\ln 3
t
>
ln
3
d)
ln
2
<
t
<
ln
3
\ln 2<t<\ln 3
ln
2
<
t
<
ln
3
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