I would like to maximize the function: 1 2 ∑ i = 1 N | x i − 1 N |under the constrains ∑ i = 1 N x i = 1 and ∀ i ∈ ( 1 , . . . , N )I have done some test for small values of N and I have the feeling that the solution is 1 − 1 N but I can't figure out how to solve it analytically.

Question

I would like to maximize the function:      1    2        ∑          i      =      1              N        |      x    i    −      1    N    |under the constrains       ∑          i      =      1              N            x    i    =  1 and   ∀  i  ∈  (  1  ,  .  .  .  ,  N  )I have done some test for small values of   N and I have the feeling that the solution is   1  −      1    N   but I can&#039;t figure out how to solve it analytically.

Kaylie Mcdonald · Accepted Answer

Look at the point in which the function takes the maximum value. If there are two numbers that lie on the different sides of   1      /    N, i.e.       x    i    &amp;lt;  1      /    N  &amp;lt;      x    j   (otherwise you can make       x    i   a little bit smaller and       x    j   the same little bit larger so that the function value will grow) then either       x    i    =  0 or       x    j    =  1. Lets assume that none of the numbers is equal to one.Now look at all nonzero numbers. They are either all greater then   1      /    N all smaller. The latter is impossible. So all numbers are greater then   1      /    N. But then the value of your function is   1  −  m      /    N where   m is the number of nonzero       x    i  s.

I would like to maximize the function: 1 2 </mfrac> <munderover> &#x221

Answered question

Answer & Explanation

New Questions in High school geometry