Problem
ALG-B2-M02-P004 One variable twice
#4
★★★☆☆ Level 3 of 5
If \(x,y>0\) and \(x^2y=64\), prove \(2x+y\ge12\).
Hint 1. Apply AM-GM to \(x,x,y\).
Hint 2. The product of the three chosen terms is known.
\(\frac{x+x+y}{3}\ge\sqrt[3]{x^2y}=4\). Therefore \(2x+y\ge12\). Equality occurs at \(x=y=4\).
AM-GM module training problem. Method tags: weighted-am-gm, fixed-product.