Minimum Efficient Scale (MES) and economic efficiency

The Minimum Efficient Scale (MES)  is the smallest quantity where the long-run average cost is lowest. It determines whether a firm has reached economies of its scale, and it is important to determine the market structure – if MES is small relative to the market demand, then the market is likely to have many firms and intense competition; on the other hand, if MES is big relative to the market demand, the market is likely to have a small number of firms. However,  it alone cannot determine whether the firm's profit is maximized or efficiency is achieved. 

To determine whether the firm's decision is optimal (in the sense of profit maximization), we need to compare whether MR is the same as MC at the current level production. 

To discuss efficiency, we always need to compare the benefit vs. the cost of an action -- looking at cost alone cannot determine whether it is efficient. A firm can produce MES, but if no consumers want the products, then the production is a waste of resources and a loss of efficiency. To determine whether the production is efficient for society, we need to compare whether the marginal social benefit (MSB) is the same as the marginal social cost (MSC). Because in this question no externalities exist, social benefit/cost is the same as private benefit/cost, we can simply compare MB and MC. Hope this makes sense.

Production efficiency indicates a producer produces a certain output in the LEAST costly way. Note that the cost function indicates the LOWEST possible cost for producing the corresponding output, so any production following the cost curve achieves production efficiency. MES is therefore not related to production efficiency.