Find the function s ( x ) such that s ( x ) maximizes ∫ 0 s − 1 ( k ) s ( x ) d xwhere x ∈ [ 0 , 10 ], s ( x ) ∈ [ 0 , 1 ], and k ∈ [ 0 , 1 ] ( k is a constant).

Question

Find the function   s  (  x  ) such that   s  (  x  ) maximizes      ∫    0                  s                  −          1                    (      k      )        s  (  x  )  d  xwhere   x  ∈  [  0  ,  10  ],   s  (  x  )  ∈  [  0  ,  1  ], and   k  ∈  [  0  ,  1  ] (  k is a constant).

Ordettyreomqu · Accepted Answer

It seems to me that, if   s  (  x  ) is strictly increasing then for   x between 0 and       s          −      1        (  k  ), we have   0  &amp;lt;  s  (  x  )  &amp;lt;  k.Therefore:      ∫    0                  s                  −          1                    (      k      )        s  (  x  )  d  x  &amp;lt;      ∫    0                  s                  −          1                    (      k      )        k  d  x  =  k  ∗      s          −      1        (  k  )  &amp;lt;=  k  ∗  10let       s    n    (  x  ), for   n  ∈  {  1  ,  2  ,  .  .  } be a sequence of functions with:      s    n    (  x  )  =  k  +  (  x  −  10  )      /    nThen       ∫    0                  s                  −          1                    (      k      )        s  (  x  )  d  x goes to 10*k when n goes to infinity.So we should say there is no such strictly increasing s(x). But we can say that sup of this integral should be 10*k.

Find the function s ( x ) such that s ( x ) maximizes <msubsup>

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