A Simple Way to Prove This Inequality | Olympiad Math

In this video, I'll show you a simple way to prove this inequality. First, we derive the inequality for the arithmetic and geometric means. Then we discuss how to apply this inequality to complete the proof.

[benjaminvatovez]

Thank you for the video.
As every term of the left side is positive, I suggest using the quadratic mean:
(Left side)/3 <= thus Left side <= (21)^(1/2) < 5 since 21 < 5².
By the way, we reach the maximum (21)^(1/2) if and only if a=b=c=1/3 which is indeed positive.

[ranshen]

Function f(x)=√(4x+1) is strictly concave for x>0. So by Jensen's inequality with uniform discrete probabiliy mass of 1/3 on each of {a, b, c},
LHS / 3 ≤ √ ( 4(a+b+c)/3 +1 ),
implying LHS ≤ √21, with equality holding iff a=b=c=1/3.