If the total cost for producing xx units of a particular product is given by C(x)C(x), then the average cost of production those xx units is given by A(x)=C(x)xA(x)=C(x)x

We will first find the average cost function. The formula to find the average cost function has already been given.

A(x)=C(x)xA(x)=15,000+100x+2x32x=15000x−1+100+2x0.5A(x)=C(x)xA(x)=15,000+100x+2x32x=15000x−1+100+2x0.5

To find the number of units at which the average cost is minimized, we have to equate the derivative of the average cost function to zero and solve for x.

A(x)=15000x−1+100+2x0.5A′(x)=−15000x−2+x−0.5A′(x)=0⇒15000x−2=x−0.5Squaring both the sides,(15000x−2)2=x−1150002x3=1x=608.22≈608unitsA(x)=15000x−1+100+2x0.5A′(x)=−15000x−2+x−0.5A′(x)=0⇒15000x−2=x−0.5Squaring both the sides,(15000x−2)2=x−1150002x3=1x=608.22≈608units

Thus, the average cost is minimized when 608 units are produced.

b)

The minimum average cost can be found by substituting the value of x as 608 in the average cost function.

A(x)=15000∗608−1+100+2∗6080.5=73.986