Problem
ALG-B2-M05-P014 Exponential and zero sum
#14
★★★★☆ Level 4 of 5
Let \(x+y+z=0\). Prove \[e^x+e^y+e^z\ge3.\]
Hint. The function \(e^x\) is convex.
By Jensen, \[\frac{e^x+e^y+e^z}{3}\ge e^{(x+y+z)/3}=e^0=1.\] Therefore the sum is at least \(3\).
Shows that a zero sum can be the average of the arguments.