No, I don't think your approach to (i) works. You have not assumed that f is differentiable. However, f is a continuous function on a closed interval. What do you know about such functions? Incidentally, I think there's an error in the problem statement. It should have
As for (ii), for the first inequality, I would experiment with (constant) functions like
You may have to take square roots, or squares, or something to get that to work with the 2-norm.
For (ii) on the second inequality, again, you're dealing with continuous functions on closed intervals. Just think about a constant function. Is there a constant function that might satisfy that inequality?