Question

The total cost function of a firm is `C = x^2 + 75x + 1600` for output x. Find the output (x) for which average
cost is minimum. Is `C_A = C_M` at this output?

Answer 1

Given cost function 

`C(x) = x^2 + 75x + 1600` 

Average   `bar C (x)=(C(x))/x` 

=`(x^2+75x+1600)/x`

=`x+75+1600/x` 

Now `barC'(x)=(dbarC(x))/dx=1-1600/x^2` 

For minimum average cost `barC (x)=0` 
∴Minimum average cost=`barC(x)=40+75+1600/40=155` 

∴ `C_A=155` 

Now we find marginal cost i.e., 

`C_m=(dC)/(Dx)` 

C_m=`d/dx(x^2+75x+1600)` 

= 2x + 75               ...(1) 

∴ put x=40 in eq (1) 

`C_m=2xx40+75` 

= `80+75=155` 

`C_A=C_m  for x=40`