Problem
ALG-B2-M10-P011 Roots with reserve
#11
★★★★★ Level 5 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[\sqrt{a}+\sqrt{b}+\sqrt{c}\le3+\frac12\left((a-1)^2+(b-1)^2+(c-1)^2\right).\]
Hint. First prove the stronger estimate \(\sum\sqrt{a}\le3\).
By Jensen for the concave function \(\sqrt{x}\), \(\frac13\sum\sqrt{a}\le\sqrt{(a+b+c)/3}=1\). Hence \(\sum\sqrt{a}\le3\). The right side of the problem is at least \(3\), since the added sum of squares is nonnegative.
Teaching goal: choose a combination of methods and always check the equality case.