| why is the function [|sinx|+|cosx|] defined only in the interval [1,root2] |
That does not look correct. Domain should be R. If the question is about range then it should not say defined in the interval but should say lies in the interval instead.
use R formula to get root 2. for the lower limit 1 this is the way:
[|sinx|+|cosx|]2 = 1 + 2|sin 2x|
from here you see that |sinx|+|cosx| = (1 + 2|sin 2x|)0.5
now since |sin 2x|>= 0 minimum value is 1. =D