We are given a ratio: g ( x ) f ( x ) where: g ( x ) ∈ R + f ( x ) ∈ N ∩ f ( x ) ≥ 2So g ( x ) returns values in [ 0 , + ∞ ] while f ( x ) returns values in { 2 , 3 , 4 , … }.I am looking for a confirmation about a very simple question: if I maximize g ( x ) f ( x ) , do I also maximize g ( x ) f ( x ) − 1 in this very particular case?

Question

We are given a ratio:            g      (      x      )              f      (      x      )      where:  g  (  x  )  ∈            R              +        f  (  x  )  ∈      N      ∩  f  (  x  )  ≥  2So   g  (  x  ) returns values in   [  0  ,  +  ∞  ] while   f  (  x  ) returns values in   {  2  ,  3  ,  4  ,  …  }.I am looking for a confirmation about a very simple question: if I maximize                     g        (        x        )                    f        (        x        )            , do I also maximize                     g        (        x        )                    f        (        x        )        −        1             in this very particular case?

Tamia Padilla · Accepted Answer

Of course you do not.An example is   g  (  x  )  =      x    2    −  100  x  +  10000, and   f  (  x  )  =  max  {  ⌈      x    ⌉  ,  2  }  .. [Sorry for not finding a nice example]In one case you have minimum is somewhere around 76-77, in the other one 79-80.EDIT: I see you wanted to maximize, I did minimize. But the reason why it doesn&#039;t hold is the same.

ntaraxq · Accepted Answer

If I interpret your question correctly, this depends on   f  (  x  ) and   g  (  x  ), e.g.:if   f  (  x  ) is constant then             g      (      x      )              f      (      x      )        =  c  ⋅  g  (  x  ) (c being a constant), so it is maximized at the same   x (obviously).if e.g.   g  (  x  ) is constant and   f  (  x  ) is not and there is a       x    0   such that   f  (      x    0    )  −  1  =  f  (  x  ), then clearly(well, by substitution) this       x    0   is the maximum. Example:       1          ⌈      x      +      2      ⌉       is maximal for   x  =  −  1 whereas       1          ⌈      x      +      2      ⌉      −      1       is maximal for   x  =  0. (  ⌈  ⋅  ⌉ is the ceiling function that returns the next larger integer for any real number)Also,             g      (      x      )              f      (      x      )      −      1       is not defined if   f  (  x  )  =  1.

We are given a ratio: g ( x ) </mrow> f

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