Problem
ALG-B2-M05-P008 Two weights for the root
#8
★★★☆☆ Level 3 of 5
Let \(x,y\ge0\). Prove \[3\sqrt{\frac{x+2y}{3}}\ge \sqrt{x}+2\sqrt{y}.\]
Hint. Use the concavity of \(\sqrt{x}\) with weights \(\frac13\) and \(\frac23\).
By the concavity of \(\sqrt{x}\), \[\sqrt{\frac{x+2y}{3}}\ge\frac13\sqrt{x}+\frac23\sqrt{y}.\] Multiply by \(3\).
Checks that the student is not tied only to equal weights.