Problem
ALG-B2-M02-P007 Weighted normalization
#7
★★★★☆ Level 4 of 5
Let \(x,y,z>0\) and \(x^2yz=16\). Prove \(2x+y+z\ge8\).
Hint 1. Take \(x,x,y,z\).
Hint 2. The product of these four numbers is \(16\).
By AM-GM for \(x,x,y,z\): \(\frac{x+x+y+z}{4}\ge\sqrt[4]{x^2yz}=2\). Hence \(2x+y+z\ge8\).
AM-GM module training problem. Method tags: weighted-am-gm, normalisation.