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Problem
Problem
Let
u
=
c
+
1
−
c
u = \sqrt{c+1} - \sqrt{c}
u
=
c
+
1
−
c
and
v
=
c
−
c
−
1
v = \sqrt{c} - \sqrt{c-1}
v
=
c
−
c
−
1
,
c
>
1
c > 1
c
>
1
and let
f
(
x
)
=
ln
(
1
+
x
)
f(x) = \ln(1+x)
f
(
x
)
=
ln
(
1
+
x
)
,
∀
x
∈
(
−
1
,
∞
)
\forall x \in (-1, \infty)
∀
x
∈
(
−
1
,
∞
)
.
Statement I
f
(
u
)
>
f
(
v
)
f(u) > f(v)
f
(
u
)
>
f
(
v
)
,
∀
c
>
1
\forall c > 1
∀
c
>
1
because
Statement II
f
(
x
)
f(x)
f
(
x
)
is increasing function, hence for
u
>
v
u > v
u
>
v
,
f
(
u
)
>
f
(
v
)
f(u) > f(v)
f
(
u
)
>
f
(
v
)
.
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