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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?
Solution
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`
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