The average cost function for x units of a commodity is A C = 1 3 x 2 + 3 x - 7 + 16 x . Show that the marginal average cost is given by x ( M C ) - C ( x ) x 2 .

Given, average cost function A C = 1 3 x 2 + 3 x - 7 + 16 x Average cost = Total cost x Hence, total cost C ( x ) = ( 1 3 x 2 + 3 x - 7 + 16 x ) × x = 1 3 x 3 + 3 x 2 - 7 x + 16 Marginal cost function M C = d d x ( C ( x ) ) M C = d d x ( 1 3 x 3 + 3 x 2 - 7 x + 16 ) ⇒ M C = x 2 + 6 x - 7 Marginal average cost = d d x ( A C ) = d d x ( x 2 3 + 3 x - 7 + 16 x ) = 2 3 x + 3 - 16 x 2 x ( M C ) - C ( x ) x 2 = x ( x 2 + 6 x - 7 ) - ( x 3 3 + 3 x 2 - 7 x + 16 ) x 2 = x 3 + 6 x 2 - 7 x - x 3 3 - 3 x 2 + 7 x - 16 x 2 = 2 x 3 3 + 3 x 2 - 16 x 2 = 2 3 x + 3 - 16 x 2 Hence, marginal average cost = x ( M C ) - C ( x ) x 2

The average cost function for x units of a commodity is A C = 1 3 x 2 + 3 x - 7 + 16 x . Show that the marginal average cost is given by x ( M C ) - C ( x ) x 2 .

Given, average cost function A C = 1 3 x 2 + 3 x - 7 + 16 x Average cost = Total cost x Hence, total cost C ( x ) = ( 1 3 x 2 + 3 x - 7 + 16 x ) × x = 1 3 x 3 + 3 x 2 - 7 x + 16 Marginal cost function M C = d d x ( C ( x ) ) M C = d d x ( 1 3 x 3 + 3 x 2 - 7 x + 16 ) ⇒ M C = x 2 + 6 x - 7 Marginal average cost = d d x ( A C ) = d d x ( x 2 3 + 3 x - 7 + 16 x ) = 2 3 x + 3 - 16 x 2 x ( M C ) - C ( x ) x 2 = x ( x 2 + 6 x - 7 ) - ( x 3 3 + 3 x 2 - 7 x + 16 ) x 2 = x 3 + 6 x 2 - 7 x - x 3 3 - 3 x 2 + 7 x - 16 x 2 = 2 x 3 3 + 3 x 2 - 16 x 2 = 2 3 x + 3 - 16 x 2 Hence, marginal average cost = x ( M C ) - C ( x ) x 2

Verify that d d x ( A C ) = ( M C - A C ) x for C ( x ) = a x + b + c

Given, cost function C ( x ) = a x + b + c Average cost A C = Total cost x Hence, A C = a x + b + c x Marginal cost function M C = d d x ( C ( x ) ) M C = d d x ( a x + b + c ) ⇒ M C = d d x ( a x + b ) + d d x ( c ) ⇒ M C = d d x ( a x + b ) ⇒ M C = a 2 a x + b Marginal average cost = d d x ( A C ) = d d x ( a x + b + c x ) = d d x ( a x + b x ) + d d x ( c x ) = d d x ( a x + b x ) - c x 2 = a 2 x a x + b - a x + b c x - c x 2 ( M C - A C ) x = 1 x [ a 2 a x + b - a x + b c - c x ] = a 2 x a x + b - a x + b c x - c x 2 Hence, d d x ( A C ) = ( M C - A C ) x

Verify that d d x ( A C ) = ( M C - A C ) x for C ( x ) = 5 x 2 x 2 + 3 + 600

Given, cost function C ( x ) = 5 x 2 x 2 + 3 + 600 Average cost A C = Total cost x Hence, A C = C ( x ) = 5 x 2 x 2 + 3 + 600 x = C ( x ) = 5 x x 2 + 3 + 600 x Marginal cost function M C = d d x ( C ( x ) ) M C = d d x ( 5 x 2 x 2 + 3 + 600 ) ⇒ M C = x 2 + 3 ( 10 x ) - 5 x 2 1 2 x 2 + 3 2 x x 2 + 3 ⇒ M C = 5 x 3 + 30 x x 2 + 3 3 2 Marginal average cost = d d x ( A C ) = d d x ( 5 x x 2 + 3 + 600 x ) = x 2 + 3 · 5 - 5 x · 1 2 x 2 + 3 · 2 x x 2 + 3 - 600 x 2 = 15 x 2 + 3 3 2 - 600 x 2 ( M C - A C ) x = 1 x [ 5 x 3 + 30 x x 2 + 3 3 2 - 5 x x 2 + 3 - 600 x ] Simplifying ( M C - A C ) x = = 15 x 2 + 3 3 2 - 600 x 2 Hence, d d x ( A C ) = ( M C - A C ) x

The average cost function A C for a commodity is given by A C = x + 5 + 36 x in terms of output x . Find the total cost and the marginal cost as the functions of x .

Given, A C = x + 5 + 36 x Total cost = Average cost × Number of output units Total cost C ( x ) = ( x + 5 + 36 x ) x = x 2 + 5 x + 36 Marginal cost is rate of change of total cost. Marginal cost M C = d d x ( C ) = d d x ( x 2 + 5 x + 6 ) = 2 x + 5 Hence, total cost is x 2 + 5 x + 36 and 2 x + 5 .

The average cost function AC for a commodity is given by A C = x + 5 + 36 x in terms of output x. Find the output for which AC increases.

Given, average cost function A C = x + 5 + 36 x To check output for which AC increases, we will differentiate A C d d x ( A C ) = d d x ( x + 5 + 36 x ) = 1 - 36 x 2 For A C to be increasing, d d x ( A C ) > 0 1 - 36 x 2 > 0 1 > 36 x 2 x 2 > 36 x > 6 Hence, for x > 6 , A C increases.