# Maximizing $\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}$ over $a+b+c=4abc$ and $a,b,c>0$

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?

• 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? Feb 28 at 17:41