If the short-run cost curve of a firm is U-shaped, the marginal and average cost are equal where the
average variable cost is minimum
marginal cost is falling
average cost is minimum
average fixed cost is falling
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The point of intersection blw marginal and average cost is also the minimum of the AC cost too

The diagram below shows the AFC, AVC, ATC, and Marginal Costs (MC) curves:
It is important to note that the behaviour of the ATC curve depends upon that of the AVC and AFC curves. Observe that:
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.

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.

