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1
inequality with powers
inequality with powers
Source:
November 25, 2005
inequalities
calculus
derivative
function
induction
inequalities proposed
Problem Statement
Prove the following inequality:
x
1
+
2
x
2
+
3
x
3
+
.
.
.
+
n
x
n
≤
n
(
n
−
1
)
2
+
x
1
+
x
2
2
+
x
3
3
+
.
.
.
+
x
n
n
x_1 + 2x_2 + 3x_3 + ... + nx_n \leq \frac{n(n-1)}{2} + x_1 + x_2 ^2 + x_3 ^3 + ... + x_n ^n
x
1
+
2
x
2
+
3
x
3
+
...
+
n
x
n
≤
2
n
(
n
−
1
)
+
x
1
+
x
2
2
+
x
3
3
+
...
+
x
n
n
where
∀
x
i
x
i
>
0
\forall _{x_i} x_i > 0
∀
x
i
x
i
>
0
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