A few weeks ago I took part in a math competition and had this question through some trial and error

res2bfitjq

res2bfitjq

Answered question

2022-05-28

A few weeks ago I took part in a math competition and had this question through some trial and error I got the correct answer 80 as I didn't have time to use maximization using calculus, is there a way to do this question quickly in a rigorous way:
Find the maximum possible value of
9 x + 8 y + 5 z
where x, y, and z are positive real numbers satisfying
9 x + 4 y + z = 128
Any help would be greatly appreciated, thanks.

Answer & Explanation

mseralge

mseralge

Beginner2022-05-29Added 10 answers

Rename ( x , y , z ) to ( x 1 , x 2 , x 3 ), and let ( c 1 , c 2 , c 3 ) = ( 9 , 8 , 5 ) and ( a 1 , a 2 , a 3 ) = ( 9 , 4 , 1 ). By Cauchy-Schwarz:
( j c j x j ) 2 = ( j c j a j a j x j ) 2 ( j ( c j a j ) 2 ) ( j a j x j 2 ) = ( j c j 2 a j ) ( j a j x j ) = ( 9 2 9 + 8 2 4 + 5 2 1 ) 128 = 6400 ,
so
j c j x j 80

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