Returns to a single variable factor, also known as returns to a factor, refers to the change in output resulting from increasing one input (like labor or capital) while keeping all other inputs constant. This concept is often illustrated through the law of diminishing marginal returns, which states that as more of one input is added, with all other inputs held constant, the additional output gained from the extra input will eventually decrease. On the other hand, returns to scale refers to the change in output resulting from a proportional increase in all inputs. If output increases by the same proportion as the increase in inputs, we have constant returns to scale. If output increases by a greater proportion, we have increasing returns to scale. If output increases by a lesser proportion, we have decreasing returns to scale. 

Now, let's analyze the cost-minimizing condition for the firm. The condition for optimizing the usage of inputs in production is that the ratio of the marginal product of labor (MPL) to the wage rate (w) should equal the ratio of the marginal product of capital (MPK) to the rental rate of capital (r). In mathematical terms, this is MPL/w = MPK/r. In