 # 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 $G\left(k,x\right)={\int }_{a}^{x}f\left(k,t\right)dt$, and I wish to maximize $G\left(k,x\right)$ w.r.t. $k$. Under what conditions is this maximization problem equivalent to maximizing $f\left(k,t\right)$ w.r.t. $k$?

In short, when is the following true:
$\underset{k}{max}\left\{{\int }_{a}^{x}f\left(k,t\right)\right\}dt⇔\underset{k}{max}\left\{f\left(k,t\right)\right\}$
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Do you mean to find the argmax of $k$? Assuming nice differentiability and solution is interior, the solution is given by $\frac{\mathrm{d}}{\mathrm{d}k}G\left(k,x\right)={\int }_{a}^{x}\frac{\mathrm{d}}{\mathrm{d}k}f\left(k,t\right)\mathrm{d}t=0$. You see immediately that if $k$ is the $\mathrm{arg}maxf\left(k,t\right)$ for all $t\in \left(a,x\right)$ then your claim is true.

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