Greatest Displacement and Greatest Speed

On a closed time interval, a greatest or least value may occur at an endpoint. For displacement, compare s at the endpoints and every admissible stationary time v=0. For greatest distance from the origin, compare |s| instead; for distance from the starting point, compare |s-s(0)|. These are different quantities. For maximum speed, compare |v| at the endpoints and at interior candidates where a=0, also checking any joins or points where the formula is not differentiable. Rest times v=0 are possible minimum-speed points. The greatest velocity need not give the greatest speed. For illustration, let v=2-t for 0 <= t 5. The greatest velocity is 2 at the start, but the greatest speed is 3 at the end. The particle rests at t=2, where its direction changes. To decide whether speed is increasing, examine the signs of velocity and acceleration: equal nonzero signs mean increasing speed; opposite signs mean decreasing speed. Negative acceleration can increase speed when velocity is negative. At a zero or join, examine the neighbouring intervals rather than relying on one product value.