Problem
ALG-B2-M05-P009 Maximum sum of roots
#9
★★★☆☆ Level 3 of 5
Let \(x,y,z\ge0\) and \(x+y+z=48\). Find the maximum possible value of \(\sqrt{x}+\sqrt{y}+\sqrt{z}\).
Hint. For a concave function, the maximum occurs at equal arguments.
By Jensen, \[\frac{\sqrt{x}+\sqrt{y}+\sqrt{z}}{3}\le\sqrt{\frac{48}{3}}=4.\] Hence the sum is at most \(12\). Equality occurs at \(x=y=z=16\), so the maximum is \(12\).
This is an extremum problem, not only an inequality proof.