Given a constant sum of x n values: ∑ n = 1 N x n = Cwhere x n ≥ 1Find the maximum of the following expression: ∑ n = 1 N a n log ⁡ ( x n ) where the a n &gt; 0 values are constants, and the x n values are free to change, given that their sum will remain constant.

Question

Given a constant sum of       x    n   values:      ∑          n      =      1              N                                          x          n                            =  Cwhere       x    n    ≥  1Find the maximum of the following expression:      ∑          n      =      1              N                                          a          n                    log      ⁡                        (                                    x              n                                )                          where the       a    n    &amp;gt;  0 values are constants, and the       x    n   values are free to change, given that their sum will remain constant.

Ashly Sanford · Accepted Answer

Calling       y    n    =      x    n    −  1 we have the equivalent problem  max      ∑          n      =      1        N        a    n    ln  ⁡  (      y    n    +  1  )        s. t.               ∑          n      =      1        N        y    n    =  C  −  Nso the lagrangian reads  L  (  y  ,  λ  )  =      ∑          n      =      1        N        a    n    ln  ⁡  (      y    n    +  1  )  −  λ      (          ∑              n        =        1            N              y      n        −    C    +    N    )  The stationary points are the solutions for  ∇  L  =  0  =      {                                                      a              n                                                      y                n                            +              1                                                =                          λ                                                  ∑                          n              =              1                        N                                y            n                                    =                          C          −          N                        substituting      y    n    =            a      n        λ    −  1into the restriction we have      1    λ        ∑          n      =      1        N        a    n    −  N  =  C  −  Nor  λ  =      1    C        ∑          n      =      1        N        a    n    ⇒      y    n    =                    a        n            C                      ∑                  n          =          1                N                    a        n              −  1  ⇒      x    n    =                    a        n            C                      ∑                  n          =          1                N                    a        n            expecting of course that       a    k    ,  C are such that                     a        n            C                      ∑                  n          =          1                N                    a        n              ≥  1. Here, as log is an strict increasing function, there exist C such that this positive condition is verified.

Find the maximum of the following expression: sum_(n=1)^N a_n log(x_n) where the a_n>0 values are constants

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New Questions in Calculus 1