2. The cost in dollars of producing $\mathrm{x}$ items is $C(x)=4 x^{2}+676$ and the revenue from the sale of the $\mathrm{x}$ items is $R(x)=12 x^{2}+200$. a) (5 pts) Find the $(x, y)$ coordinates of the average cost when the slope of the tangent line is horizontal. Also, graph the average cost function over an appropriate window and indicate your window settings. b) ( 3 pts) When $x=4$ items, evaluate the average profit. Use 2 decimal place accuracy as needed.

Step 1 ：The average cost function is given by \(A(x) = \frac{C(x)}{x} = 4x + \frac{676}{x}\).

Step 2 ：To find the x-coordinate where the slope of the tangent line is horizontal, we need to find where the derivative of the average cost function is zero.

Step 3 ：The derivative of the average cost function is \(A'(x) = 4 - \frac{676}{x^2}\).

Step 4 ：Setting this equal to zero and solving for x gives: \(4 - \frac{676}{x^2} = 0\)

Step 5 ：\(4 = \frac{676}{x^2}\)

Step 6 ：\(x^2 = \frac{676}{4}\)

Step 7 ：\(x^2 = 169\)

Step 8 ：\(x = \sqrt{169} = 13\)

Step 9 ：So the x-coordinate where the slope of the tangent line is horizontal is \(x = 13\).

Step 10 ：To find the y-coordinate, we substitute \(x = 13\) into the average cost function:

Step 11 ：\(A(13) = 4*13 + \frac{676}{13} = 52 + 52 = 104\)

Step 12 ：So the (x, y) coordinates of the average cost when the slope of the tangent line is horizontal are \(\boxed{(13, 104)}\).

Step 13 ：The profit function is given by \(P(x) = R(x) - C(x) = 12x^2 + 200 - 4x^2 - 676 = 8x^2 - 476\).

Step 14 ：The average profit when \(x = 4\) is then \(\frac{P(4)}{4} = \frac{8*4^2 - 476}{4} = \frac{128 - 476}{4} = -87\).

Step 15 ：So when \(x = 4\) items, the average profit is \(\boxed{-\$87.00}\).