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Mathematics
Mathematics
Problem
If
f
(
x
)
f(x)
f
(
x
)
is positive for all positive values of
x
x
x
and
f
′
(
x
)
<
0
f'(x) < 0
f
′
(
x
)
<
0
,
f
′
′
(
x
)
>
0
f''(x) > 0
f
′′
(
x
)
>
0
,
∀
x
∈
R
+
\forall x \in \mathbb{R}^+
∀
x
∈
R
+
, prove that
1
2
f
(
1
)
+
∫
1
n
f
(
x
)
d
x
<
∑
r
=
1
n
f
(
r
)
<
∫
1
n
f
(
x
)
d
x
+
f
(
1
)
.
\frac{1}{2} f(1) + \int_{1}^{n} f(x)\,dx < \sum_{r=1}^{n} f(r) < \int_{1}^{n} f(x)\,dx + f(1).
2
1
f
(
1
)
+
∫
1
n
f
(
x
)
d
x
<
r
=
1
∑
n
f
(
r
)
<
∫
1
n
f
(
x
)
d
x
+
f
(
1
)
.
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