Mathematics
Mathematics
Problem
Explain why determining the greatest distance a particle travels on a closed time interval requires checking both the times when the derivative of its position is zero and the endpoints of the interval.Intuitive explanation
The key insight is that extrema of a continuous function on a closed interval can occur in two places: where the derivative equals zero (critical points, where the particle momentarily stops changing direction) or at the boundaries themselves. A particle might reach its farthest point while actively moving, right at the moment it stops—or it might have already been at maximum distance at the start or end of your observation window. Checking only critical points misses these endpoint possibilities, while checking only endpoints misses interior turning points.
Intuitive explanation
The key insight is that extrema of a continuous function on a closed interval can occur in two places: where the derivative equals zero (critical points, where the particle momentarily stops changing direction) or at the boundaries themselves. A particle might reach its farthest point while actively moving, right at the moment it stops—or it might have already been at maximum distance at the start or end of your observation window. Checking only critical points misses these endpoint possibilities, while checking only endpoints misses interior turning points.