Yes, you can choose any interval, but the one you choose must have different signs for the values at the ends of the interval.
And yes, the max and min, if they exist, are values of the original function.
Want to just clear two minor things up:
If they ask me to use IVT to prove there is a pos/neg root of a function, and they don't give an interval, I can choose any interval, right?
Also, if they ask me to find local max/min of function I plug the critical numbers into the ORIGINAL function, right?