The selling price of a commodity is fixed at Rs . 60 and its cost function is C x = 35 x + 250 . Find the break even points.

Given cost function is C x = 35 x + 250 and selling price is Rs . 60 . Here revenue R x = 60 x ∴ Profit function P x = R x - C x = 60 x - 35 x + 250 ⇒ P x = 25 x - 250 For break even points, P x = 0 ⇒ 25 x - 250 = 0 ⇒ 25 x = 250 ⇒ x = 10 Hence, break even point is x = 10 .

The selling price of a commodity is fixed at Rs . 60 and its cost function is C x = 35 x + 250 . Determine the profit function.

Given cost function is C x = 35 x + 250 and selling price is Rs . 60 . Here revenue R x = 60 x ∴ Profit function P x = R x - C x = 60 x - 35 x + 250 ⇒ P x = 25 x - 250 Hence, profit function is 25 x - 250 .

The total cost function for a production is given by C ( x ) = 3 4 x 2 - 7 x + 27 . Find the number of units produced for which M . C . = A . C . ( M . C . = Marginal Cost and A . C . = Average Cost.)

Total cost function for a production is given by C ( x ) = 3 4 x 2 - 7 x + 27 And, Marginal cost M . C . = d d x C x = d d x 3 4 x 2 - 7 x + 27 = 3 4 d d x x 2 - 7 d d x x + d d x 27 = 3 4 2 x - 7 + 0 ∵ d d x x n = n x n - 1 , n ∈ R = 3 2 x - 7 . . . . . i Now, we know that Average cost A . C . = Total cost Number of units of a good produced ⇒ A . C . = 3 4 x 2 - 7 x + 27 x . . . . i i Now, according to the question, using i , and i i , we have M . C . = A . C . ⇒ 3 2 x - 7 = 3 4 x 2 - 7 x + 27 x ⇒ x 3 2 x - 7 = 3 4 x 2 - 7 x + 27 ⇒ 3 2 x 2 - 7 x = 3 4 x 2 - 7 x + 27 ⇒ 3 4 x 2 = 27 ⇒ x 2 = 27 3 × 4 = 36 ⇒ x = 6 Hence, the number of units produced is 6 .

The marginal cost function of x units of a product is given by MC = 3 x 2 - 10 x + 3 . The cost of producing one unit is ∓ 7 . Find the total cost function and average cost function.

Given, M C = 3 x 2 - 10 x + 3 ∴ C ( x ) = ∫ 3 x 2 - 10 x + 3 d x = x 3 - 5 x 2 + 3 x + C When x = 1 , C ( 1 ) = 7 ∴ 7 = ( 1 ) 3 - 5 ( 1 ) 2 + 3 ( 1 ) + C ⇒ C = 7 - 1 + 5 - 3 = 8 Therefore, C ( x ) = x 3 - 5 x 2 + 3 x + 8 Average cost function = C ( x ) x = x 2 - 5 x + 3 + 8 x

The cost function of a product is given by C ( x ) = x 3 3 - 45 x 2 - 900 x + 36 where x is the number of units produced. How many units should be produced to minimise the marginal cost?

Given, C ( x ) = x 3 3 - 45 x 2 - 900 x + 36 Let f ( x ) be the marginal function, then f ( x ) = d d x C ( x ) = x 2 - 90 x - 900 f ' ( x ) = 2 x - 90 and f ' ' ( x ) = 2 For marginal cost to be minimum, we must have f ' ( x ) = 0 ⇒ 2 x - 90 = 0 ⇒ x = 45 At x = 45 , f ' ' ( x ) = 2 > 0 Hence, the marginal cost is minimum, when number of units produced is 45 .