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, $$ \bar{C}(x) $$. $$ x=21 $$ (a) Find the instantaneous rate of change of average cost with respect to the number of units produced, at any level of production. (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.

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Let's work through each part of the problem step by step.

### Given:
- **Total Cost Function:** $$ C(x) = 4,410 + 85x + 0.1x^2 $$

### (a) Find the Instantaneous Rate of Change of the Average Cost Function

**Average Cost Function ($$ \bar{C}(x) $$):**
$$
\bar{C}(x) = \frac{C(x)}{x} = \frac{4,410 + 85x + 0.1x^2}{x} = \frac{4,410}{x} + 85 + 0.1x
$$

**Instantaneous Rate of Change ($$ \frac{d\bar{C}}{dx} $$):**
$$
\frac{d\bar{C}}{dx} = \frac{d}{dx}\left( \frac{4,410}{x} + 85 + 0.1x \right) = -\frac{4,410}{x^2} + 0 + 0.1 = 0.1 - \frac{4,410}{x^2}
$$

### (b) Find the Level of Production Where the Rate of Change Equals Zero

Set the instantaneous rate of change to zero and solve for $$ x $$:
$$
0.1 - \frac{4,410}{x^2} = 0
$$
$$
\frac{4,410}{x^2} = 0.1
$$
$$
x^2 = \frac{4,410}{0.1} = 44,100
$$
$$
x = \sqrt{44,100} = 210
$$

**Thus, the level of production is $$ x = 210 $$ units.**

### (c) Evaluate the Statement at $$ x = 210 $$

**Statement:** *At the value found in part (b), the average cost function is greater than the total cost function.*

**Analysis:**
$$
\bar{C}(x) = \frac{C(x)}{x}
$$
For $$ \bar{C}(x) > C(x) $$, it would imply:
$$
\frac{C(x)}{x} > C(x) \Rightarrow 1 > x
$$
However, since $$ x = 210 $$, which is greater than 1, the inequality $$ \bar{C}(x) > C(x) $$ **does not hold**.

**Conclusion:** The statement is **false**. In reality, for $$ x > 1 $$, the average cost $$ \bar{C}(x) $$ is less than the total cost $$ C(x) $$.

### Summary of Answers:
- **(a)** The instantaneous rate of change is $$ \frac{d\bar{C}}{dx} = 0.1 - \frac{4,410}{x^2} $$.
- **(b)** The level of production is $$ x = 210 $$ units.
- **(c)** The statement "**The average cost function is greater than the total cost function**" is **false** at $$ x = 210 $$.

Answer:
Problem (a) Answer

The instantaneous rate of change of average cost is $$ \frac{d\bar{C}}{dx} = 0.1 - \frac{4,\!410}{x^{2}} $$.
The instantaneous rate of change of average cost is $$ \frac{d\bar{C}}{dx} = 0.1 - \frac{4,410}{x^{2}} $$.

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