Velocity extrema and speed extrema

Velocity is signed; speed is its magnitude. An absolute maximum or minimum on a closed time interval must be compared with endpoint values. Why: a=dv/dt identifies interior stationary velocity candidates. But |v| also has a minimum wherever v=0, and a large negative velocity may have the largest magnitude. Method: find zeros of acceleration inside the interval; justify their nature using its sign or the second derivative of velocity; evaluate velocity at these points and both endpoints. Compare signed values for velocity and absolute values for speed; include all zeros of velocity for minimum speed. For v=(t-1)2-2 on 0 <= t 3, velocities at the candidates t=0,1,3 are -1,-2,2. Least velocity is -2; greatest speed is 2 at both 1,3. Least speed is zero at t=1+ 2, the only zero in the interval. Check: a=0 alone proves neither a maximum nor a minimum. Endpoints can supply the greatest speed even when no stationary velocity exists.