Fermi
·
Concept Library
·
Courses
·
Books
·
Problems
·
Simulations
·
Math Wiki
·
High School
Fermi
›
Problems
Problem
Problem
If
f
:
(
0
,
∞
)
→
R
f: (0, \infty) \to R
f
:
(
0
,
∞
)
→
R
be given by
f
(
x
)
=
∫
1
/
x
x
e
−
(
t
+
1
t
)
d
t
t
f(x) = \int_{1/x}^{x} e^{-\left(t+\frac{1}{t}\right)} \frac{dt}{t}
f
(
x
)
=
∫
1/
x
x
e
−
(
t
+
t
1
)
t
d
t
, then, a.
f
(
x
)
f(x)
f
(
x
)
is monotonically increasing on
[
1
,
∞
)
[1, \infty)
[
1
,
∞
)
b.
f
(
x
)
f(x)
f
(
x
)
is monotonically decreasing on
[
0
,
1
)
[0, 1)
[
0
,
1
)
c.
f
(
x
)
+
f
(
1
x
)
=
0
,
∀
x
∈
(
0
,
∞
)
f(x) + f\left(\frac{1}{x}\right) = 0, \forall x \in (0, \infty)
f
(
x
)
+
f
(
x
1
)
=
0
,
∀
x
∈
(
0
,
∞
)
d.
f
(
2
x
)
f(2^x)
f
(
2
x
)
is an odd function of
x
x
x
on
R
R
R
Browse the problem library