Question

The cost function of a firm is C = x³ – 12x² + 48x. Find the level of output (x > 0) at which average cost is minimum.

Answer 1

The cost function is C = x³ – 12x² + 48x

Average cost is minimum,

When Average Cost (AC) = Marginal Cost (MC)

Cost function, C = x³ – 12x² + 48x

Average Cost, AC = `(x^3 - 12x^2 + 48x)/x` = x² – 12x + 48

Marginal Cost (MC) = `"dC"/"dx"`

`= "d"/"dx" (x^3 - 12x^2 + 48x)`

= 3x² – 24x + 48

But AC = MC

x² – 12x + 48 = 3x² – 24x + 48

x² – 3x² – 12x + 24x = 0

-2x² + 12x = 0

Divide by -2 we get, x² – 6x = 0

x (x – 6) = 0

x = 0 (or) x – 6 = 0

x = 0 (or) x = 6

But x > 0

∴ x = 6

Output = 6 units