When you are working out dy/dx = 0, why do you do this and what does it mean?

It relates back to what we actually mean why we say dy/dx. Simply put, dy/dx means the rate of change of y with respect to the rate of change in x over a infinitely small space of time. Therefore, when we are saying dy/dx is equal to zero, we are saying that the rate of change in the y axis is 0 with respect to the x axis, in other words y is not changing. (I would then put a curve of say y=x^2 to a student and ask them to see if they can work out where dy/dx = 0 is (minimum) and why it was there). Whenever you are saying dy/dx = 0, the curve will be instantly flat, known as stationary points. When looking at other things, this idea of dy/dx can be really useful. For example, if you are to throw a ball in the air, you can work out where the ball is at its highest as dx/dt (velocity) is zero if you just have how the ball changes position with respect to time.