Suppose that the total cost function, in dollars, for the production of \( x \) units of a product is given by the equation shown below. \( C(x)=4,410+85 x+0.1 x^{2} \) Find the equation of the average cost function, \( \bar{C}(x) \). (a) Find the instantaneous rate of change of average cost with respect to the number of units produced, at any level of production. 210 (b) Find the level of production at which this rate of change equals zero. \( x=210 \) (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.

Let's analyze part (c) of your problem step-by-step to determine the correctness of the given statement. ### Recap of Previous Parts 1. **Total Cost Function:** \[ C(x) = 4,\!410 + 85x + 0.1x^2 \] 2. **Average Cost Function:** \[ \bar{C}(x) = \frac{C(x)}{x} = \frac{4,\!410}{x} + 85 + 0.1x \] 3. **Instantaneous Rate of Change of Average Cost (First Derivative):** \[ \bar{C}'(x) = -\frac{4,\!410}{x^2} + 0.1 \] 4. **Critical Point (Setting \(\bar{C}'(x) = 0\)):** \[ -\frac{4,\!410}{x^2} + 0.1 = 0 \implies x^2 = \frac{4,\!410}{0.1} = 44,\!100 \implies x = 210 \] ### Analyzing Part (c) **Statement:** > "The average cost function is greater than the total cost function." Let's evaluate this at \( x = 210 \). 1. **Calculate Total Cost at \( x = 210 \):** \[ C(210) = 4,\!410 + 85(210) + 0.1(210)^2 = 4,\!410 + 17,\!850 + 4,\!410 = 26,\!670 \text{ dollars} \] 2. **Calculate Average Cost at \( x = 210 \):** \[ \bar{C}(210) = \frac{4,\!410}{210} + 85 + 0.1(210) = 21 + 85 + 21 = 127 \text{ dollars} \] 3. **Compare \( \bar{C}(210) \) and \( C(210) \):** \[ \bar{C}(210) = 127 \text{ dollars} \\ C(210) = 26,\!670 \text{ dollars} \] Clearly, \( \bar{C}(210) < C(210) \). ### Conclusion The statement "**The average cost function is greater than the total cost function**" is **false** at \( x = 210 \). In reality, the average cost per unit is **less than** the total cost at this level of production. **Correct Interpretation:** - At \( x = 210 \), the average cost function reaches its minimum value. - The average cost per unit (\( \bar{C}(x) \)) is significantly less than the total cost (\( C(x) \)). ### Final Answer **False. At x = 210, the average cost is less than the total cost, so the statement is not true.**