Question

The Average Cost of producing ‘x’ units of commodity is given by:

`AC = x^2/200 - x/50 - 30 + 5000/x`

1. Find the Cost function.   [1]
2. Find the Marginal Cost function.   [1]
3. Find the Marginal Average Cost function.   [1]
4. Verify that `d/dx (AC) = (MC - AC)/x`   [1]

Answer 1

Given, Average cost (AC)

= `x^2/200 - x/50 - 30 + 5000/x`

a. Cost function (CT)

= x × Average cost

= `x(x^2/200 - x/50 - 30 + 5000/x)`

`CT = x^3/200 - x^2/50 - 30x + 5000`

b. Marginal cost function

= `(d(CT))/dx`

= `d/dx (x^3/200 - x^2/50 - 30x + 5000)`

= `(3x^2)/200 - (2x)/50 - 30`

`MC = (3x^2)/200 - x/25 - 30`

c. Marginal Average Cost

= `d/dx` (Average cost)

= `d/dx (x^2/200 - x/50 - 30 + 5000/x)`

`MAC = (2x)/200 - 1/50 - 5000/x^2`

`MAC = x/100 - 1/50 - 5000/x^2`

d. L.H.S. `d/dx (AC) = x/100 - 1/50 - 5000/x^2`

R.H.S. `(MC - AC)/x = 1/x [((3x^2)/200 - x/25 - 30) - (x^2/200 - x/50 - 30 + 5000/x)]`

= `1/x [(2x^2)/200 - x/25 + x/50 - 5000/x]`

= `1/x [x^2/100 - x/50 - 5000/x]`

= `x/100 - 1/50 - 5000/x^2`

= `d/dx (AC)`

Here, L.H.S. = R.H.S.

`d/dx (AC) = (MC - AC)/x`

Hence Proved.