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Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 4abc.$ Find the maximum value of $$\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}.$$


First, I tried using $\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}=\dfrac{4}{\sqrt{bc}}+\dfrac{6}{\sqrt{ac}}+\dfrac{12}{\sqrt{ab}}$. Then, I tried to apply some inequalities, but it didn't work.

I also tried just substituting $abc=\dfrac{a+b+c}{4}$ into the equation, but this also didn't do much. Could someone give me some guidance on how to proceed?

Thanks in advance!!!!

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  • $\begingroup$ Use lagrange multipliers $\endgroup$ Feb 28 at 17:22
  • $\begingroup$ And, what inequalities did you try? $\endgroup$ Feb 28 at 17:22
  • $\begingroup$ Hmm... What are Lagrange multipliers? I tried using inequalities like QM-AM. $\endgroup$ Feb 28 at 17:24
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    $\begingroup$ One thing that should make it substantially easier: divide both sides of the constraint by $abc$ to get $\frac1{bc}+\frac1{ac}+\frac1{ab}=4$. Now you can do a change of variables $x=\frac1{\sqrt{bc}}$, $y=\frac1{\sqrt{ac}}$, $z=\frac1{\sqrt{ab}}$. (First prove that we can recover $a, b, c$ from $x, y, z$ so this is a one-to-one mapping!) What form does the constraint take in terms of this $x,y,z$? What does the function to optimize look like? Do you know how to optimize that sort of function subject to that constraint? $\endgroup$ Feb 28 at 17:41
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    $\begingroup$ @Steven Stadnicki Yes, thanks. Just easy Cauchy. :) $\endgroup$ Feb 28 at 17:59

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