Explanation
The long-run average total cost is U-shaped owing to the economies and the diseconomies of the scale. Economies of scale are defined as the cost advantages that occur when an increase in the output produced lowers the average total cost. On the other hand, the diseconomies of scale are the cost disadvantages that occur when the firm experiences rising average total costs upon increasing the output produced.

In the initial stages of production, the firm experiences factor specialization and use efficient capital owing to which the average total cost keeps on falling with an increase in the production. This in turn, makes the firm realize the economies of scale. Later on, as the plant size increases and the output produced becomes too large, there arises management complexity, which in turn raises the average total costs. This makes the firm realize the diseconomies of scale in the later period. The falling average total cost initially and the rising of the same later on make the long-run average total cost curve U shaped.

The minimum efficient scale can be defined as the lowest level of output required to be produced so as make the firm move down to the minimum point of the average total cost curve where all the economies of scale would be realized.

The shape of the long-run average total cost describes the market structure and the concentration of small and large firms operating in the market.

In case, the long-run ATC curve has a relatively lower downward sloping segment implying that it quicks falls to the minimum point, and then stretches over a large output range, then there would be both large and few firms in the market.
In case, the long-run ATC curve has a relatively larger downward sloping segment implying that it slowly approaches the minimum point, then there would exist large firms in the market that are only a few in number.
In case, the long-run ATC curve approaches its minimum point in quick succession and thereafter witnesses a sudden rise, then there would be many small firms in the market.
Verified Answer
The U-shape of the long-run average total cost is because of the economies of scale in the initial production stage and diseconomies of scale in the later stages. When the firm experiences economies of scale, the average cost declines as the output produced rises. On the other hand, when the firm experiences the diseconomies of scale, the average cost rises with rising output.

The lowest level of output required to make the firm produce at the minimum point of the long-run ATC curve and extract all the economies of scale is the minimum efficient scale.

There exist both large and small firms in the market if the long-run ATC curve reaches the minimum point quickly and thereafter stretches itself over a larger output range. On the other hand, there would be few large firms operating in case the long-run ATC slowly approaches the minimum point. There would be many small firms in the market if the long-run ATC drops the minimum point quickly and thereby rises suddenly.