Either Best or waste
Determine the greatest value of the sum M = x y + y z + z x where x , y , and z are real numbers satisfying the following condition x 2 + 2 y 2 + 5 z 2 = 2 2 .
The answer is 11.
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2 solutions
Thnx bro,NICE solution
We have that x 2 + y 2 + z 2 > = x y + y z + z x or simply ( x − y ) 2 + ( y − z ) 2 + ( z − x ) 2 > = 0
Transforming the given condition we get: ( x − y ) 2 + ( y − z ) 2 + ( z − x ) 2 + 2 M − x 2 now by inequality that expression is bigger than 2 M − x 2 or 2 2 > = 2 M − x 2 now we just minimize x and get the wanted result 11
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Thanks for the short solution.
And for the latex; brackets are not entered correctly. Please check.
Here we go:
We have
( x − y − z ) 2 + ( y − 2 z ) 2 ≥ 0 ∀ x , y , z ∈ R
⇔ x 2 + 2 y 2 + 5 z 2 − 2 x y − 2 y z − 2 x z ≥ 0
⇔ x 2 + 2 y 2 + 5 z 2 ≥ 2 x y + 2 y z + 2 x z
⇔ 2 ( x y + y z + x z ) ≤ 2 2
⇔ x y + y z + x z ≤ 1 1
So, Max ( x y + y z + x z ) = 1 1 .
I want more good solutions. So, please post them.