The cost function of a firm is C = x3 – 12x2 + 48x. Find the level o

1.	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» 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 = \(\frac{x^3-12x^2+48x}{x}\) = x² – 12x + 48 Marginal Cost (MC) = \(\frac{dC}{dx}\) = \(\frac{d}{dx}\)(x³ – 12x² + 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

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»

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 = \(\frac{x^3-12x^2+48x}{x}\) = x² – 12x + 48

Marginal Cost (MC) = \(\frac{dC}{dx}\)

= \(\frac{d}{dx}\)(x³ – 12x² + 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