Problem
ALG-B2-M05-P001 Mean of squares
#1
★★☆☆☆ Level 2 of 5
Prove for all real \(a,b,c\): \[a^2+b^2+c^2\ge\frac{(a+b+c)^2}{3}.\]
Hint. Take \(f(x)=x^2\).
The function \(x^2\) is convex. Jensen gives \(\frac{a^2+b^2+c^2}{3}\ge\left(\frac{a+b+c}{3}\right)^2\). Multiplying by \(3\) gives the result.
Basic problem on the direction of Jensen for a convex function.