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Serbia Team Selection Test
1976 Yugoslav Team Selection Test
Problem 3
min/max with four variables over R+0
min/max with four variables over R+0
Source: Yugoslav TST 1976 P3
May 29, 2021
inequalities
Problem Statement
Find the minimum and maximum values of the function
f
(
x
,
y
,
z
,
t
)
=
a
x
2
+
b
y
2
a
x
+
b
y
+
a
z
2
+
b
t
2
a
z
+
b
t
,
(
a
>
0
,
b
>
0
)
,
f(x,y,z,t)=\frac{ax^2+by^2}{ax+by}+\frac{az^2+bt^2}{az+bt},~(a>0,b>0),
f
(
x
,
y
,
z
,
t
)
=
a
x
+
b
y
a
x
2
+
b
y
2
+
a
z
+
b
t
a
z
2
+
b
t
2
,
(
a
>
0
,
b
>
0
)
,
given that
x
+
z
=
y
+
t
=
1
x+z=y+t=1
x
+
z
=
y
+
t
=
1
, and
x
,
y
,
z
,
t
≥
0
x,y,z,t\ge0
x
,
y
,
z
,
t
≥
0
.
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