Problem
ALG-B2-M05-P019 Fourth power with fixed sum
#19
★★★★★ Level 5 of 5
Let \(x,y,z\ge0\) and \(x+y+z=3\). Prove \[x^4+y^4+z^4\ge3.\]
Hint. Use the convexity of \(t^4\), or the tangent to \(t^4\) at \(1\).
By Jensen, \[\frac{x^4+y^4+z^4}{3}\ge\left(\frac{x+y+z}{3}\right)^4=1.\] Thus the sum of fourth powers is at least \(3\). Equality holds at \(x=y=z=1\).
The form is simple, but the idea is important: strong convexity gives a minimum at equal variables.