Suppose that c(x)=3x3−6x2+11,000xc(x)=3x3−6x2+11,000x is the cost of manufacturing xx items

The given cost function is:

c(x)=3x3−6x2+11000xc(x)=3x3−6x2+11000x

Finding the average cost function by using the formula:

A(x)=c(x)xA(x)=3x3−6x2+11000xxA(x)=3x2−6x+11000A(x)=c(x)xA(x)=3x3−6x2+11000xxA(x)=3x2−6x+11000

Finding the derivative of A(x) with respect to x, we get:

A′(x)=6x−6A′(x)=6x−6

Putting A'(x) = 0, we get:

6x−6=0x=16x−6=0x=1

Finding the second derivative of A(x), we get:

A′′(x)=6>0A″(x)=6>0

Implies that, A(x) is minimum at x = 6.

The production level that minimizes the average cost of making xx items is x=6x=6.