Problem
ALG-B2-M09-P007 Maximum sum of roots
#7
★★★☆☆ Level 3 of 5
Let \(x,y,z\ge0\), \(x+y+z=12\). Find the maximum of \(\sqrt{x}+\sqrt{y}+\sqrt{z}\).
Hint. The square root is concave.
By Jensen, \(\frac{\sqrt{x}+\sqrt{y}+\sqrt{z}}3\le\sqrt{\frac{x+y+z}{3}}=2\). Thus the sum is at most \(6\). Equality holds at \(x=y=z=4\).
Concavity gives a maximum at equality.