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

Define f(x)f(x) as the average cost function, i.e. f(x)=C(x)xf(x)=C(x)x. The average cost is minimized when f′(x)=0f′(x)=0. We calculate this as follows:

f(x)=81,000+170x+6x32xf′(x)=ddx(81,000+170x+6x32x)=x⋅ddx(81,000+170x+6x32)−(81,000+170x+6x32)ddx(x)x2use quotient rule=x(170+6⋅32x12)−(81,000+170x+6x32)(1)x2use power rule=170x+9x32−81,000−170x−6x32x2=3x32−81,000x2f′(x)=03x32−81,000x2=03x32−81,000=03x32=81,000x32=27,000(x32)23=(27,000)23x=(303)23=302=90f(x)=81,000+170x+6x32xf′(x)=ddx(81,000+170x+6x32x)=x⋅ddx(81,000+170x+6x32)−(81,000+170x+6x32)ddx(x)x2use quotient rule=x(170+6⋅32x12)−(81,000+170x+6x32)(1)x2use power rule=170x+9x32−81,000−170x−6x32x2=3x32−81,000x2f′(x)=03x32−81,000x2=03x32−81,000=03x32=81,000x32=27,000(x32)23=(27,000)23x=(303)23=302=90

Therefore, the average cost is minimized when 9090 units are produced.