= – . dy dx < - - These points divides the whole interval into three intervals namely (–∞,– ),(– , ) and ( ,∞). Fig : . -∞ ∞ - Interval Sign of ¢ f ( x ) Intervals of increasing/ decreasing (–∞, – ) (–) (–) > Increasing in (–∞, – ] (– , ) (–) (+) < Decreasing in [– , ] ( , ∞) (+) (+) > Increasing in [ , ∞) Table: .
Example . Find the stationary value and the stationary points f ( x )= x + x – . Given that f ( x ) = x + x – … ( ) f' ( x ) = x + At stationary points, ¢ f ( x ) = Þ x + = Þ x = – f ( x ) has stationary value at x = – When x = – , from ( ) f (– ) = (– ) + (– )– = – Stationary value of f ( x ) is – Hence stationary point is (– ,– ) Example . Find the stationary values and stationary points for the function: f ( x )= x + x + x + .
Solution : Given that f ( x ) = x + x + x + . f' ( x ) = x + x + = ( x + x + ) = ( x + )( x + ) f' ( x ) = Þ ( x + )( x + ) = Þ x + = (or) x + = . Þ x = – (or) x = – f ( x ) has stationary points at x = – and x = – Stationary values are obtained by putting x = – and x = – . When x = – , f (– )= (– )+ ( )+ (– )+ = – When x = – , f (– )= (– )+ ( )+ (– )+ =– The stationary points are (– ,– ) and (– ,– ).
Example . The profit function of firm in producing x units of a product is given