Monotonic means if it increases, it does not also decrease. If it decreases, it does not also increase. Since it increases, it can be monotonic if and only if it does not decrease (which is true). The derivative can be approximated by taking the slope of the lines between adjacent points. Those slopes would be 1,2,3. So, it is possible that f'(x) is increasing. Next, II and III are saying the exact same thing. If II is true, then so is III (and vise versa). For III, you can only discount something if you have evidence to the contrary. Since you do not know if the function exists for x<-0.5 or x>-0.2, you cannot discount that it is possible. There is no evidence to the contrary, so it is possible.