Problem
ALG-B2-M05-P002 Sum of square roots
#2
★★☆☆☆ Level 2 of 5
Let \(x,y,z\ge0\) and \(x+y+z=27\). Prove \[\sqrt{x}+\sqrt{y}+\sqrt{z}\le9.\]
Hint. The function \(\sqrt{x}\) is concave.
By Jensen for the concave function \(\sqrt{x}\), \[\frac{\sqrt{x}+\sqrt{y}+\sqrt{z}}{3}\le\sqrt{\frac{x+y+z}{3}}=\sqrt{9}=3.\] Hence the sum of roots is at most \(9\).
Stress the direction: the square root is concave, so it gives an upper bound.