can study the trend of the business related to the function and therefore, we can predict or forecast the business trend. Example . Show that the function f ( x ) = x x x ∈ is strictly increasing function on R . Solution : f ( x ) = x x x ∈ f '( x ) = x – x + = x – x + + = ( x – ) + > , for all x Î R Therefore, the function f is strictly increasing on (-∞,∞).
Example . Find the interval in which the function f ( x )= x – x + is strictly increasing and strictly decreasing. Given that f ( x ) = x – x + Differentiate with respect to x , ¢ f ( x ) = x – When ¢ f ( x ) = Þ x – = Þ x = . Then the real line is divided into two intervals namely (–∞, ) and ( ,∞) Fig : .
-∞ ∞ [To choose the sign of ¢ f ( x ) choose any values for x from the intervals ans substitute in ¢ f ( x ) and get the sign.] Interval Sign of ′ ( ) = Nature of the function (-∞, ) < f ( x ) is strictly decreasing in (-∞, ) ( , ∞) > f ( x ) is strictly increasing in ( , ∞) Table: . Example . Find the intervals in which the function f given by f ( x )= x – x – x + is increasing or decreasing. Solution : f ( x ) = x – x – x + ¢ f ( x ) = x – x – = ( x – x – ) = ( x – )( x + ) ¢ f ( x ) = Þ ( x – )( x + )= x = (or) x = – f ( x ) has stationary at x = and at x