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Let \(a,b,c,d\)   be positive real numbers. Find the minimum value of \(\frac{(a + b)(a + c)(b + c)}{abc}. \)
 

I think you're supposed to use AM-GM inequalities, but I have no idea how. Help!

 Apr 20, 2020
 #1
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The minimum value is 4.

 Apr 21, 2020
 #2
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Let \(a,b,c,d\) be positive real numbers.

Find the minimum value of \(\dfrac{(a + b)(a + c)(b + c)}{abc}\)

 

\(\mathbf{\huge{AM \geq GM }}\)

 

\(\begin{array}{|rcll|} \hline \dfrac{a+b}{2} & \ge & \sqrt{ab} \\\\ \dfrac{a+c}{2} & \ge & \sqrt{ac} \\\\ \dfrac{b+c}{2} & \ge & \sqrt{bc} \\\\ \hline \left( \dfrac{a+b}{2} \right) \left( \dfrac{a+c}{2} \right) \left( \dfrac{b+c}{2} \right) & \ge & \sqrt{ab}\sqrt{ac} \sqrt{bc} \\\\ \dfrac{(a + b)(a + c)(b + c)}{8} & \ge & \sqrt{abacbc} \\\\ \dfrac{(a + b)(a + c)(b + c)}{8} & \ge & \sqrt{a^2b^2c^2} \\\\ \dfrac{(a + b)(a + c)(b + c)}{8} & \ge & abc \\\\ \dfrac{(a + b)(a + c)(b + c)}{abc} & \ge & 8 \\\\ \mathbf{ 8 } & \le & \mathbf{\dfrac{(a + b)(a + c)(b + c)}{abc}} \\ \hline \end{array}\)

 

The minimum value is 8

 

laugh

 Apr 21, 2020
 #3
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thanks so much, heureka!!!! it was correct!

Guest Apr 23, 2020

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