For the smallest possible value of AE, A or E must be in the middle of the line. Consider A (E if you want) is that point. With AB = 3 and BC = 6, with A at the middle, then AC = 3. With CD = 8, AC = 3 and A at the middle, then AD = 5. With DE = 4, AD = 5 so D is in the middle, then AE = 1 (the smallest possible value of AE)

This can be more simply illustrated using Vevtors and magnitude of vectors since they all lie on a line.