Problem
ALG-B2-M09-P010 Maximum squares on the boundary
#10
★★★☆☆ Level 3 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Find the maximum of \(a^2+b^2+c^2\).
Hint. Use \(a^2\le3a\), \(b^2\le3b\), \(c^2\le3c\).
Since \(0\le a,b,c\le3\), we have \(a^2\le3a\), \(b^2\le3b\), \(c^2\le3c\). Therefore \(\sum a^2\le3(a+b+c)=9\). Equality occurs at \((3,0,0)\) and permutations.
Contrasts with the minimum of squares: the maximum is on the boundary.