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 = 128Any help would be greatly appreciated, thanks.

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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&#039;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  =  128Any help would be greatly appreciated, thanks.

mseralge · Accepted Answer

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

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

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