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South East Mathematical Olympiad
2009 South East Mathematical Olympiad
7
Find minimum value and maximum value of f(x,y,z)
Find minimum value and maximum value of f(x,y,z)
Source:
September 18, 2010
inequalities
inequalities proposed
Problem Statement
Let
x
,
y
,
z
≥
0
x,y,z\geq0
x
,
y
,
z
≥
0
be real numbers such that
x
+
y
+
z
=
1
x+y+z=1
x
+
y
+
z
=
1
Define
f
(
x
,
y
,
z
)
f(x,y,z)
f
(
x
,
y
,
z
)
in this way :
f
(
x
,
y
,
z
)
=
x
(
2
y
−
z
)
1
+
x
+
3
y
+
y
(
2
z
−
x
)
1
+
y
+
3
z
+
z
(
2
x
−
y
)
1
+
z
+
3
x
f(x,y,z)=\frac{x(2y-z)}{1+x+3y}+\frac{y(2z-x)}{1+y+3z}+\frac{z(2x-y)}{1+z+3x}
f
(
x
,
y
,
z
)
=
1
+
x
+
3
y
x
(
2
y
−
z
)
+
1
+
y
+
3
z
y
(
2
z
−
x
)
+
1
+
z
+
3
x
z
(
2
x
−
y
)
Find the minimum value and maximum value of
f
(
x
,
y
,
z
)
f(x,y,z)
f
(
x
,
y
,
z
)
.
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