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

The average cost function is given as:

C(x)x=81000+170x+6x32xC(x)x=81000+170x+6x32x

Now, we will find the derivative of this cost function to get the critical point, as shown below:

ddx(81000+170x+6x32x)=ddx(81000+170x+6x32)x−ddx(x)(81000+170x+6x32)x2            [?(fg)′=f′⋅g−g′⋅fg2]=(9x12+170)x−1⋅(81000+170x+6x32)x2                [?ddx(xa)=a⋅xa−1]=3x32−81000x2ddx(81000+170x+6x32x)=ddx(81000+170x+6x32)x−ddx(x)(81000+170x+6x32)x2            [?(fg)′=f′⋅g−g′⋅fg2]=(9x12+170)x−1⋅(81000+170x+6x32)x2                [?ddx(xa)=a⋅xa−1]=3x32−81000x2

Now, the critical point is when:

ddx(81000+170x+6x32x)=0⇒3x32−81000x2=0⇒x=3√729000000ddx(81000+170x+6x32x)=0⇒3x32−81000x2=0⇒x=7290000003

So, at this production level, the average cost is minimum.