I have a simple question. This might be a theorem somewhere, but I do not know the appropriate keywo

EnvivyEvoxys6 2022-07-02 Answered
I have a simple question. This might be a theorem somewhere, but I do not know the appropriate keywords to find it. Please help.

Say there is a function G ( k , x ) = a x f ( k , t ) d t, and I wish to maximize G ( k , x ) w.r.t. k. Under what conditions is this maximization problem equivalent to maximizing f ( k , t ) w.r.t. k?

In short, when is the following true:
max k { a x f ( k , t ) } d t max k { f ( k , t ) }
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Answers (1)

amanhantmk
Answered 2022-07-03 Author has 17 answers
Do you mean to find the argmax of k? Assuming nice differentiability and solution is interior, the solution is given by d d k G ( k , x ) = a x d d k f ( k , t ) d t = 0. You see immediately that if k is the arg max f ( k , t ) for all t ( a , x ) then your claim is true.

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