Suppose that the total cost function, in dollars, for the production of \( x \) units of a product is given by the equation shown below. Find the equation of the average cost function, \( \bar{C}(x) \). \( \bar{C}(x)=410+85 x+0.1 x^{2} \) \( x=21 \) (a) Find the instantaneous rate of change of average cost with respect to the number of units produced, at any level of production. \( x=3+x \) (b) Find the level of production at which this rate of change equals zero. \( x=21 \) (c) At the value found in part (b), which of the following is true? The average cost function is greater than the total cost function. The tangent line to the average cost function is a horizontal line.

To find the equation of the average cost function \( \bar{C}(x) \) based on the total cost function provided, divide the total cost \( C(x) = 410 + 85x + 0.1x^2 \) by \( x \). This gives \( \bar{C}(x) = \frac{C(x)}{x} = \frac{410}{x} + 85 + 0.1x \). Now, for the instantaneous rate of change, you'll need to differentiate \( \bar{C}(x) \) with respect to \( x \). The derivative, \( \bar{C}'(x) \), can be found as follows: \( \bar{C}'(x) = -\frac{410}{x^2} + 0.1 \). Next, setting \( \bar{C}'(x) = 0 \) leads us to find the level of production where the rate of change is zero. Solving \( -\frac{410}{x^2} + 0.1 = 0 \) will give the necessary \( x \). When \( x \) equals the value found in part (b), you’ll be looking at where the average and total costs intersect. If \( \bar{C}(x) \) is evaluated at this production level, it indicates that the average cost continues to decrease beyond this point, and the tangent line to \( \bar{C}(x) \) at this point will indeed be horizontal, indicating an inflection point. In simpler terms, it's the sweet spot in production where costs level off before they start to rise again! You recognize the balance between production and expenditure, which is crucial in cost management. For those curious about how this applies in the real world, consider how companies routinely analyze their production costs to determine optimal levels of output. By understanding their average cost functions, businesses can make informed decisions about pricing, production rates, and scaling strategies. To dive deeper, there are numerous resources on cost functions in economic theory and production efficiency. Books and online courses on microeconomics often include sections on cost functions, helping you get a grip on how businesses gauge their performance and pricing in the competitive market!