In the beginning, both AVC and AFC curves fall. Hence, the ATC curve falls as well.

Next, the AVC curve starts rising, but the AFC curve is still falling. Hence, the ATC curve continues to fall. This is because,during this phase, the fall in the AFC curve is greater than the rise in the AVC curve.

As the output rises further, the AVC curve rises sharply. This offsets the fall in the AFC curve. Hence, the ATC curve falls initially and then rises.
If the average cost falls due to an increase in the output, the marginal cost is less than the average cost.

If the average cost rises due to an increase in the output, the marginal cost is more than the average cost.

Marginal cost is equal to the average cost when the marginal cost is minimum. You can see in Fig. 1 that the MC curve cuts the ATC curve at its minimum or optimum point.