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!
+26400 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
