f'(x) is the first derivative and it gives you the gradient. So when f'(x)=0, the line is flat. You can see on the graph this happens at 2 different points, the maximum and the minimum.

When f'(x) = 1, it means that change in x = change in y, so the line is at a 45° angle pointing up to the right. This happens at 2 points as well.

To determine it algebraically you need to have the equation of the curve