Minima . . Problems on profit maximization and minimization of cost function: Example . For a particular process, the cost function is given by C , where C is cost per unit and x , the number of unit’s produced.
Find the minimum value of the cost and the corresponding number of units to be produced. Differentiate with respect to x , dC =- and dx d C dC & - ⇒ x = When x = , dx d C ` C is minimum when x = The minimum value of cost = – + = The corresponding number of units produced = . Example . The total cost function of a firm is C x ^ h , where x is the output.
A tax at the rate of ` per unit of output is imposed and the producer adds it to his cost. If the market demand function is given by p = – x , where p is the price per unit of output, find the profit maximizing the output and price. Total revenue: R = p x = ( – x ) x = – Tax at the rate ` per x unit = x . - - Applications of Differentiation ` C ( x ) + x = x P = Total revenue – (Total cost + tax ) = ( – x x dP = and dx d P dP = & = Þ x = ` x = (– is not acceptable) At x = dx d P = - < P is maximum when x = .
` P = – ( ) = ` . Example . The manufacturing cost of an item consists of ` , as over head material cost ` per item and the labour cost ` k for x items produced. Find how many items be produced to have the minimum average