Suppose that the total cost function, in dollars, for the production of \( x \) units of a product is given by the equation shown below. \[ (x)=11,250+65 x+0.5 x^{2} \] Find the equation of the average cost function, \( \bar{C}(x) \). \( \bar{C}(x)= \) (a) Find the instantaneous rate of change of average cost with respect to the number of units produced, at any level of production. (b) Find the level of production at which this rate of change equals zero. \( x= \) (c) At the value found in part (b), which of the following is true? O The average cost function is greater than the total cost function. The tangent line to the total cost function is a horizontal line. The instantaneous rate of change of total cost is zero. The tangent line to the average cost function is a horizontal line.

To find the average cost function, \( \bar{C}(x) \), we divide the total cost function \( C(x) \) by the number of units \( x \): \[ \bar{C}(x) = \frac{C(x)}{x} = \frac{11,250 + 65x + 0.5x^2}{x} = \frac{11,250}{x} + 65 + 0.5x \] Now, for part (a), the instantaneous rate of change of the average cost function with respect to \( x \) is found by differentiating \( \bar{C}(x) \): \[ \bar{C}'(x) = -\frac{11,250}{x^2} + 0.5 \] For part (b), to find when this rate equals zero, set \( \bar{C}'(x) = 0 \): \[ -\frac{11,250}{x^2} + 0.5 = 0 \implies \frac{11,250}{x^2} = 0.5 \implies 11,250 = 0.5 x^2 \implies x^2 = 22,500 \implies x = 150 \] So \( x = 150 \). For part (c), at \( x = 150 \), since we have found that the instantaneous rate of change of the average cost function is zero, this indicates that the tangent line to the average cost function is a horizontal line. Therefore, the correct statement is: "The tangent line to the average cost function is a horizontal line."