I know how to solve maximization problems on numbers, and I know how to solve differential equations which are equations on functions, but how do I solve a 'maximization problem' over functions?Here is a specific problem:Find a positive real function F ( x ), continuous and monotonically increasing on the real interval [ 0 , 1 ], which maximizes: F ( x ) F ( 1 ) Subject to: F ′ ( x ) = ( 1 − x ) F ″ ( x )what is the function F which attains this maximum?

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I know how to solve maximization problems on numbers, and I know how to solve differential equations which are equations on functions, but how do I solve a &#039;maximization problem&#039; over functions?Here is a specific problem:Find a positive real function   F  (  x  ), continuous and monotonically increasing on the real interval   [  0  ,  1  ], which maximizes:            F      (      x      )              F      (      1      )      Subject to:      F    ′    (  x  )  =  (  1  −  x  )      F    ″    (  x  )what is the function   F which attains this maximum?

svirajueh · Accepted Answer

You have a separable differential equation                    (                  F          ′                (        x        )        )            ′                      F        ′            (      x      )        =      1          1      −      x        ,i.e.  ln  ⁡      (          F      ′        (    x    )    )    =  C  −  ln  ⁡      |    1  −  x      |    ,      F    ′    (  x  )  =      C          1      −      x        ,  F  (  x  )  =  C  ln  ⁡      |    1  −  x      |    +      C    ′    .Due to the singularity at 1, I don&#039;t see a better solution than a constant.

polivijuye · Accepted Answer

For   x  &amp;lt;  1, second equation yields                    F        ″            (      x      )                      F        ′            (      x      )        =      1          1      −      x      or, integrating  ln  ⁡      F    ′    (  x  )  =  −  ln  ⁡  (  1  −  x  )  +  Cand this implies      F    ′    (  x  )  =      K          1      −      x      and integrating again,  F  (  x  )  =  J  −  K  ln  ⁡  (  1  −  x  )where   C  ,  J  ∈      R   and   K  &amp;gt;  0.Continuity of   F at   x  =  1 implies  F  (  1  )  =      lim          x      →              1        −                        (        J  −  K  ln  ⁡  (  1  −  x  )            )      but this limit is not finite, so the second equation can not be satisfied on the interval [0,1].

I know how to solve maximization problems on numbers, and I know how to solve differential equations

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