Why are both the short-run and the long-run average cost curves u-shaped?

The short-run and long-run average cost curves are U-shaped for different reasons.

For the short-run average cost curve, the initial decline is due to economies of scale. That is, as more is produced, the fixed cost is spread over more units, lowering the average fixed cost. But as more and more output is produced, the economies of scale effect is getting smaller. On the other hand, the average variable cost begins to rise due to diminishing marginal returns. Hence, the average cost will eventually rise.

There is no fixed cost in the long run. The initial decline in the long-run average cost curve is due to specialization and learning by doing, which reduces the marginal cost and average variable cost. As more and more is produced, diminishing marginal returns eventually will become the dominant factor, and average cost will rise with output.