Question

The cost, in dollars, to paint square feet of a house is modeled by the differentiable function . Which of the following is the best interpretation of the statement?

a) The cost function is decreasing.
b) The cost function is constant.
c) The cost function is increasing.
d) The cost function is undefined.

Answer 1

The cost function is increasing. Option C is correct.

Step-by-step explanation:

The best interpretation of the statement is that the cost function is increasing.

A differentiable function is one that has a continuous graph with no sharp turns or corners. When the cost function is increasing, it means that as the number of square feet to paint increases, the cost also increases. This makes sense because painting a larger area requires more paint, labor, and resources, which results in a higher cost.

Average and marginal cost curves tend to be U-shaped due to the concepts of economies and diseconomies of scale within the production process. Initially, when production levels are low, average costs decrease with increased production due to more efficient use of fixed resources, representing economies of scale. However, as production continues to increase, the cost of producing each additional unit starts to rise, leading to marginal costs that increase, signaling diseconomies of scale.

Cost functions are intertwined with these concepts; the marginal cost curve typically rises faster than the average cost curve because it represents the cost of producing one more unit at each level of production. This relationship is crucial for businesses as it affects their pricing strategies and profitability.