This is a bit of a longer problem – and could be expanded to start a nice investigation. The problem is given below:

Teacher notes:

Level of difficulty: Only HL

Syllabus knowledge required: This mixes a knowledge of graphs, ideas in formal proof and differentiation.

This is a good introduction to more formal mathematical techniques for finding upper/lower bounds for functions – and also in working with generalized functions. It would probably require longer than a usual starter. It would make a nice mini-investigation task and could be combined with using Desmos to draw some relevant graphs.