Inequality Problem
x 2 + 4 5 y 2 + 5 z 2 − x y − 4 y z − 3 2 z + 9 4 6
Let x , y and z be real numbers, find the minimum value of the expression above.
The answer is 5.
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2 solutions
y
=
x
2
+
4
5
y
2
+
5
z
2
−
x
y
−
4
y
z
−
3
2
z
+
9
4
6
⇒
y
=
(
x
−
y
y
)
2
+
(
y
−
2
z
)
2
+
(
z
−
3
1
)
2
+
5
We know
(
something
)
2
≥
0
thus,
y
=
(
x
−
y
y
)
2
+
(
y
−
2
z
)
2
+
(
z
−
3
1
)
2
+
5
≥
5
Using some manipulations we can write this in form of (x-y/2)² + (z-⅓)² + (2z-y)² + 5... Thus 5 is the minimum value