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If is given by
MATHEMATICS · Multi correct · CBSE · Mathematics · Integrals · Integrals
Problem
If
f
:
(
0
,
∞
)
→
R
f:(0,\infty)\to\mathbb{R}
f
:
(
0
,
∞
)
→
R
is 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: (1)
f
(
x
)
f(x)
f
(
x
)
is monotonically increasing on
[
1
,
∞
)
[1,\infty)
[
1
,
∞
)
(2)
f
(
x
)
f(x)
f
(
x
)
is monotonically decreasing on
(
0
,
1
)
(0,1)
(
0
,
1
)
(3)
f
(
x
)
+
f
(
1
x
)
=
0
f(x)+f\left(\frac{1}{x}\right)=0
f
(
x
)
+
f
(
x
1
)
=
0
, for all
x
∈
(
0
,
∞
)
x\in(0,\infty)
x
∈
(
0
,
∞
)
(4)
f
(
2
x
)
f(2^x)
f
(
2
x
)
is an odd function of
x
x
x
on
R
\mathbb{R}
R
Related theory
Increasing and Decreasing Functions
Curve Analysis
Accumulation Concepts
Fundamental theorem of calculus
Integrals
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