In the short run, the size of the plant is fixed whereas in the long run a firm can adjust its plant size. One of the choices in the long run will be the short run plant size. That is, a particular level of output will be the optimum choice in both periods. So, short run average cost curves must be tangent to the long run average cost curve. A long run involves several short runs. Consequently, the long run average cost curve will be the lower envelope of the short run cost curves.
The firm chooses that amount of fixed factors to minimize the average costs. In the long run marginal cost curves will consist of different parts of the short run marginal cost curves for each level of the fixed factor. The long run MC will be u-shaped and will intersect the long run average curve at the minimum.