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Question
The total cost function of a firm is C = x2 + 75x + 1600 for output x. Find the output for which the average cost ls minimum. Is CA= Cm at this output?
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Solution
Given C = x2 + 75x + 1600 ..........(1)
Marginal cost Cm = `(dC)/dx`
Differentiating (i) w.r.t.x
Cm = 2x + 75
Average cost CA = `C/x`
= x + 75 + `1600/x`
Diff (ii) w.r.t.x
`(dC_A)/(dx) = 1 + 1600 x ((-1)/x^2)`
= 1 - `1600/x^2`
= `(x^2 - 1600)/x^2`
If `(dC_A)/dx = 0 then .(x^2 - 1600)/x^2 = 0`
`x^2 - 1600 = 0
`x^2 = 1600`
x = 40 and x = -40
Differentiating `(dC_A)/dx` w.r.t.x
`(dC_A)/dx = d/dx (1 - 1600/x^2) = 0 - 1600 xx (-2x^-3)`
`((dC_A)/dx^2) _(at x = 40) = 3200/(40)^3 = 3200/64000`
= `1/20 > 0
Cm = 2x + 75
= 2(40) + 75
= 80 + 75 = 155
CA = x + 75 + `1600/40` = 155
Average cost is minimum for output = 40 .
Since, Cm at this output= 2(40) + .95 = 155
Cm = CA
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