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Inequality for a+b+c+d=4
Inequality for a+b+c+d=4
Source:
November 6, 2011
inequalities
calculus
Problem Statement
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be non-negative reals such that
a
+
b
+
c
+
d
=
4
a+b+c+d=4
a
+
b
+
c
+
d
=
4
. Prove the inequality
a
a
3
+
8
+
b
b
3
+
8
+
c
c
3
+
8
+
d
d
3
+
8
≤
4
9
\frac{a}{a^3+8}+\frac{b}{b^3+8}+\frac{c}{c^3+8}+\frac{d}{d^3+8}\le\frac{4}{9}
a
3
+
8
a
+
b
3
+
8
b
+
c
3
+
8
c
+
d
3
+
8
d
≤
9
4
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