Final answer:
To find the minimum average cost per unit for the cost function C = 0.001x^3 + 5x + 1458, we first derive the average cost function, which is 0.001x^2 + 5 + 1458/x, then find its derivative and set it to zero to solve for x.
Step-by-step explanation:
The question asks to find the number of units x that produces the minimum average cost per unit, ¯c, for the given cost function C = 0.001x^3 + 5x + 1458. To find the average cost function, we divide the total cost function by the number of units, x, which gives us ¯c(x) = C/x = (0.001x^3 + 5x + 1458) / x. Upon simplifying, we get ¯c(x) = 0.001x^2 + 5 + 1458/x. To find the minimum average cost, we would take the derivative of ¯c with respect to x and set it equal to zero, then solve for x.
To illustrate how to find this, let's consider the derivative of the average cost function:
¯c'(x) = 0.002x - 1458/x^2
Setting ¯c'(x) to zero for the minimum average cost point:
0.002x - 1458/x^2 = 0
Solving this equation would yield the value of x that minimizes average cost. However, this particular step requires algebraic manipulation and possibly applying numerical methods or graphing techniques to find the precise value.