To find the number of units x that produces the minimum average cost per unit c given the equation c = 0.001x³ + 5x + 250, we will need to analyze the function and find its minimum point.
The provided equation is a polynomial equation, and the average cost per unit is given as a function of x. Our goal is to minimize this function. Here’s how you can do it step by step:
- Set up the equation: The function given is:
- Find the derivative: To locate the minimum point, calculate the first derivative of the function with respect to x.
- Set the derivative to zero: To find critical points, set the first derivative equal to zero and solve for x.
- Analyze the behavior of the function: Since the leading coefficient of x³ is positive (0.001), the function will approach infinity as x increases indefinitely. Therefore, no local minima exists, but we can analyze the behavior of c(x) over a practical range to find a minimum average cost.
- Find the average cost over a range: You might want to compute the average cost per unit for specific values of x. For example, you could calculate:
c(x) = 0.001x³ + 5x + 250
c'(x) = 0.003x² + 5
0.003x² + 5 = 0
Since 0.003x² + 5 can never equal zero (as the term 5 ensures that it’s always positive), this means that there are no critical points that could give us a minimum based on the derivative test.
Average Cost (AC) = c(x) / x
As a strategy, you could compute this for different values of x (for example, from 1 to 100) to find a value that minimizes the average cost.
In conclusion, while the analytical method suggests no local minimum via the derivative, a numerical approach may yield a practical result within a range of units. You might employ numerical methods or use software tools to plot the average cost and visually identify the minimum point.