The daily total cost in dollars incurred by Triplett and Sons for producing [Math Processing Error]x cases of barbecue sauce is given by the function [Math Processing Error]C(x)=0

A)

The average cost is the total cost divided by the number of units produced. Therefore, the average cost function can be found by dividing the total cost function, C(x), by the number of units, x.

The average cost function will be:

[Math Processing Error]C¯(x)=0.000002x3+5x+400x=0.000002x2+5+400x−1

B)

We have to find the minimum point of the average cost function. This point can be found by equating the derivative of the average cost function to zero and solving for x. So, we will differentiate the function first and then work as follows.

[Math Processing Error]C¯′(x)=ddx0.000002x2+5+400x−1C¯′(x)=0.000004x−400x−2C¯′(x)=0⇒0.000004x−400x−2=00.000004x3=400∴x=464.159≈464units

Thus, the average cost is minimized when 464 units are produced.

C)

The minimum average cost will be where the average cost function takes its minimum value. This is when 464 units are produced. The minimum average cost can be found by substituting the value of x as 464 in the average cost function.