If the total cost function for a product is C(x)=3(x+6)3C(x)=3(x+6)3 collars, where x represents the number of hundreds of units produced, producing how many units will minimize the average cost? Find the minimum average cost per hundred units

Given

• Total cost function for a product is C(x)=3(x+6)3C(x)=3(x+6)3 where xx represents the number of hundreds of units produced.

Average Cost =3(x+6)3x=3(x+6)3x.

Average cost is minimum when derivative of average cost is zero. That is,

ddx(C(x)x)=03(3(x+6)2)x−3(x+6)3×(1)x2=03x(x+6)2−(x+6)3=0(x+6)2(3x−x−6)=0(x+6)2(2x−6)=0ddx(C(x)x)=03(3(x+6)2)x−3(x+6)3×(1)x2=03x(x+6)2−(x+6)3=0(x+6)2(3x−x−6)=0(x+6)2(2x−6)=0

We have, x≠−6x≠−6 because it can't be negative. Therefore, x=3x=3.

Therefore, the number of units that will minimize the average cost is 3×100=3003×100=300 units.

(b) Minimum average cost per hundred unit is,

C(3)3=3(3+6)33=729C(3)3=3(3+6)33=729

Therefore, minimum average cost per hundred unit is $ 729 729.