Problem
ALG-B2-M05-P013 Fourth powers
#13
★★★★☆ Level 4 of 5
Prove for \(a,b,c\ge0\): \[a^4+b^4+c^4\ge\frac{(a+b+c)^4}{27}.\]
Hint. The function \(x^4\) is convex on \(x\ge0\).
By Jensen, \[\frac{a^4+b^4+c^4}{3}\ge\left(\frac{a+b+c}{3}\right)^4.\] Multiplying by \(3\) gives \(\sum a^4\ge\frac{(a+b+c)^4}{27}\).
Prepares students for power means and stronger estimates.