The given total cost is:

C(x)=10000+5x+x29C(x)=10000+5x+x29

Where, x is the number of units.

(a)

The average cost is total cost divided by the number of units:

A(x)=C(x)x=10000+5x+x29xA(x)=10000x+5xx+x29xA(x)=10000x+5+x9A(x)=C(x)x=10000+5x+x29xA(x)=10000x+5xx+x29xA(x)=10000x+5+x9

This is the average cost function for the given total cost.

(b)

Differentiating A(x) with respect to x, we get:

A′(x)=−10000x2+19A′(x)=−10000x2+19

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

−10000x2+19=0x2−900009x2=0x2−90000=0x2=90000x=±300−10000x2+19=0x2−900009x2=0x2−90000=0x2=90000x=±300

The production cannot be negative. So, we have:

x=300x=300

Differentiating A(x) again, we get:

A′′(x)=20000x3A″(x)=20000x3

At x = 300

A′′(x)=200003003>0A″(x)=200003003>0

Implies that, A(x) is minimum at x = 300

In order to minimize the average cost the manufacturer should produce 300 items.