Final answer:
To find the minimum average cost, we determine the average cost function A(x) from C(x), simplify it, differentiate to find A'(x), and solve for A'(x) = 0 to identify the point where the average cost is minimized.
Step-by-step explanation:
To find the minimum average cost for the cost function C(x) = x³ + 173x + 281, we need to first determine the average cost function A(x) and then calculate its derivative called A'(x). The average cost function A(x) is the cost function C(x) divided by the quantity x, so A(x) = C(x) / x = x² + 173 + 281/x. To find A'(x), we use the quotient rule of differentiation or simplify A(x) and then differentiate. Let's simplify and differentiate.
A(x) = x² + 173 + 281/x
A(x) = x² + 173 + 281x⁻¹
Now we can find A'(x) by differentiating term by term:
A'(x) = 2x - 281x⁻²
A'(x) represents the rate of change of average cost concerning the quantity, and the minimum average cost occurs where A'(x) = 0, and the second derivative A''(x) is positive (indicating a local minimum).