The daily total cost (in dollars) incurred by Trappee and Sons for producing xx cases of TexaPep hot sauce is given by the following function

A)

The average cost is the cost per unit of production. Therefore, it's function can be found by dividing the total cost function, C(x), by the number of units produced, x.

¯C(x)=C(x)x=0.000002x3+4x+340x=0.000002x2+4+340x−1C¯(x)=C(x)x=0.000002x3+4x+340x=0.000002x2+4+340x−1

B)

The minimum point of the average cost function can be found by equating its derivate to zero and solving for x.

¯C′(x)=ddx0.000002x2+4+340x−1=0.000004x−340x−2¯C′(x)=0⇒0.000004x=340x−20.000004x3=340x=439.68≈440C¯′(x)=ddx0.000002x2+4+340x−1=0.000004x−340x−2C¯′(x)=0⇒0.000004x=340x−20.000004x3=340x=439.68≈440

Thus, the average cost is minimized when 440 units are produced.

C)

The marginal cost curve crosses the average cost curve at the average cost curve's minimum point. We already know that the average cost is minimum when 440 units are produced. Therefore, the average and marginal costs are equal when 440 units are produced.