During which interval is the largest amount of energy required to stretch the spring?

The further the spring is stretched from its origin.

Explanation:

When a spring is stretched or compressed from its original source, a restoring force will act, trying to bring it back to its original state. Hence, Hooke's Law states that the extension of an elastic material is proportional to the load applied or force applied to it. This will continue to hold provided the elastic limit of such material is not exceeded. Therefore, the more it is pull away from the original state, the more restoring force act on it and the more force the that will be needed to pull it away provided the elastic limit is not exceeded.