If C(x) is the cost of producing x units of a commodity, then the average cost per unit is c(x) = C(x)/x

First, we calculate the average cost function:

C(x)=54000+140x+4x32c(x)=C(x)xc(x)=54000+140x+4x32xc(x)=54000x+140+4x12C(x)=54000+140x+4x32c(x)=C(x)xc(x)=54000+140x+4x32xc(x)=54000x+140+4x12

(a) Calculating the derivative and equaling it to zero, we determine the critical points:

c(x)=54000x+140+4x12c′(x)=−54000x2+412x−12c′(x)=−54000x2+2x−12=0−54000x2+2x−12=0−54000x2=−2x−1227000=x32x=(27000)23=900c(x)=54000x+140+4x12c′(x)=−54000x2+412x−12c′(x)=−54000x2+2x−12=0−54000x2+2x−12=0−54000x2=−2x−1227000=x32x=(27000)23=900

With the second derivative test, we verify that it is a minimum:

c′(x)=−54000x2+2x−12=−54000x−2+2x−12c′′(x)=108000x−3−x−32c′′(900)>0c′(x)=−54000x2+2x−12=−54000x−2+2x−12c″(x)=108000x−3−x−32c″(900)>0

(b) Evaluating in the function, we obtain the minimum average cost:

c(900)=54000900+140+4(900)12=60+140+120=320