Question

The cost of manufacturing x units of a commodity is 27 + 12x + 3x².

Find the output for which Average Cost is decreasing.

Answer 1

Find Average Cost (AC):

Average Cost is the total cost divided by the number of units (x).

`AC = (C(x))/x = (27 + 12x + 3x^2)/x`

`AC = 27/x + 12 + 3x` 

Condition for Decreasing Average Cost:

Average Cost decreases when its derivative with respect to x is negative: `(d(AC))/dx < 0`

Differentiate AC:

`(d(AC))/dx = d/dx(27x^(−1) + 12 + 3x)`

`(d(AC))/dx = −27/x^2 + 3`

For Average Cost to be decreasing:

`3 − 27/x^2 < 0`

`3 < 27/x^2`

3x² < 27

x² < 9

x < 3