Final answer:
Marginal cost equals average total cost when ATC is at its minimum because MC intersects ATC exactly at that point, indicating the most efficient level of production.
Step-by-step explanation:
When average total cost (ATC) is at its minimum, marginal cost (MC) equals the average total cost (ATC). This relationship is due to the economic interpretation of the cost curves. If MC is below ATC, it pulls the ATC downward, indicating that production is becoming more efficient. Conversely, if MC is above ATC, it indicates that producing additional units is less efficient and thus pushes ATC upward. Therefore, the point where MC equals ATC is where ATC is at its minimum, which also suggests that the ATC curve is U-shaped and MC cuts it at its lowest point.