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June 11th, 2018, 12:07 AM  #1 
Newbie Joined: Jun 2018 From: Spain Posts: 1 Thanks: 0  Optimization Problem with a decreasing constraint
I need to find this maximum, $$ \max_{a\leq \frac{b}{1db}}\frac{1}{1+da}(a\log(a)(1a)\log(1a)) $$ where $b\rightarrow 0$, $d=\frac{n}{b}$ and $n\in\mathbb{R}$ is a constant. __________________________________________________ _________________________ Here is what I've done, first I tried to simplify the problem, so I studied the case where $b$ is still going to $0$ but $d$ is a constant. And the maximum is (obviously) archived just taking $a$ to be the higher bound. With the complete problem (where $d=\frac{n}{b}$ and $b$ is going to $0$), I started for doing some plots of specific cases, and it seems that the evaluation of the higher bound is again the maximum... (At least in the cases I saw) but of course this is just to have an idea, and it is not a proof... For the proof I tried to optimize the expression evaluated in some $xb$ (because the value of $a$ I think will depend on $b$) but I didn't get any interesting. Any help will be appreciated! 
June 11th, 2018, 12:58 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,607 Thanks: 616 
The condition $a\le \frac{b}{1db}, \ d=\frac{n}{b}$ is the same as $a\le \ \frac{b}{1n}$ which becomes $a\le 0$ in the limit as $b \to 0$. You now have a problem in that $a \lt 0$ gives a complex number, so you need to use $a=0$ and expression = 0.


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calcs, constraint, decreasing, limits, maximization, optimization, problem 
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