Final answer:
To determine if the horizontal position is a minimum or maximum, one must evaluate the second derivative of position, with a negative value indicating a relative maximum and a positive value indicating a relative minimum.
Step-by-step explanation:
To determine whether the horizontal position of the pumpkin trajectory represents a minimum or maximum, we must evaluate the second derivative of the position concerning time, d²y/dx², at xm. A negative second derivative at the horizontal position indicates a relative maximum, while a positive second derivative indicates a relative minimum. In the context of projectile motion, maximum and minimum points on a trajectory relate to the peak and the trough of the motion, respectively.
In a similar analysis concerning energy, if the second derivative of the energy with respect to position is negative, then that position is a relative maximum and indicates an unstable equilibrium. Conversely, if the second derivative is positive, the position is a relative minimum representing a stable equilibrium. This is directly related to the potential energy curve of a system.