by P ( x )= x + x + x . Check whether the firm is running a profitable business or not. P ( x ) = x + x + x . P '( x ) = x + x + =( x + ) It is clear that P '( x )> for all x .
` The firm is running a profitable business. IMPORTANT NOTE Let R ( x ) and C ( x ) are revenue function and cost function respectively when x units of commodity is produced. If R ( x ) and C ( x ) are differentiable for all x > , then P ( x ) = R ( x ) – C ( x ) is ma x imized when Marginal Revenue = Marginal cost. i.e.
when R l ( x )= C l ( x ) profit is ma x imum at its stationary point. Example . Given C ( x )= x + x + and p ( x ) = – x are the cost price and selling price when x units of commodity are produced. Find the level of the production that maximize the profit.
- - Applications of Differentiation Given C ( x ) = x + x + … ( ) and p ( x ) = – x …( ) Profit is maximized when, marginal revenue = marginal cost. (i.e) R l ( x ) = C l ( x ) = x + = p x × = x – x R '( x ) = – x Hence – x = x + x = At x = , the profit is maximum. . .
Local and Global (Absolute) Maxima and Minima