Let be t ∈ R , n = 1 , 2 , ⋯, p ∈ [ 0 , 1 ] and a ∈ ( p , 1 ].Show that sup t ( t a − log ⁡ ( p e t + ( 1 − p ) ) = a log ⁡ ( a p ) + ( 1 − a ) log ⁡ ( 1 − a 1 − p ) So, I've try to derivate, but I did'nt get sucess, since my result is different. I've get log ⁡ ( a ( 1 − p ) p ( 1 − a ) ) . Any ideas?

Question

Let be   t  ∈      R  ,   n  =  1  ,  2  ,  ⋯,   p  ∈  [  0  ,  1  ] and   a  ∈  (  p  ,  1  ].Show that      sup          t            (    t    a    −    log    ⁡    (    p          e      t        +    (    1    −    p    )    )    =  a  log  ⁡      (          a      p        )    +  (  1  −  a  )  log  ⁡      (                  1        −        a                    1        −        p              )  So, I&#039;ve try to derivate, but I did&#039;nt get sucess, since my result is different. I&#039;ve get   log  ⁡      (                  a        (        1        −        p        )                    p        (        1        −        a        )              )  . Any ideas?

Carolyn Farmer · Accepted Answer

You are given a function   f of the variable   t, with two parameters   p and   a. You are asked to verify that the supremum of the function has a certain specific form in terms of the two parameters.The correct procedure to find the supremum (or maximum) is to differentiate the function   f  (  t  ) with respect to the variable   t. This give you the first derivative. The next step is to set the first derivative equal to zero, and solve the equation in terms of   t. From your post I see that you have done so, and found that   t  =  l  o  g  (  a  )  −  l  o  g  (  1  −  a  )  +  l  o  g  (  1  −  p  )  −  l  o  g  (  p  ). This answer is correct !Now all you have to do is substitute this particular value fot   t into the expression for   f  (  t  ). This step will yield the supremum. If you perform this calculation (try it!), you will indeed find the result that was specified in the exercise.

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