is (a) increasing (b) decreasing. Solution We have f ( x ) = Therefore f ′ ( x ) = ( ( ( ) = ( )( )( ) x (on simplification) Fig . Now f ′ ( x ) = gives x = , x = – , or x = . The points x = , – , and divide the real line into four disjoint intervals namely, (– ∞ , – ), (– , ), ( , ) and ( , ∞ ) (Fig .
). Consider the interval (– ∞ , – ), i.e., when – ∞ < x < – . In this case, we have x – < , x + < and x – < . (In particular, observe that for x = – , f ′ ( x ) = ( x – ) ( x + ) ( x – ) = (– ) (– ) (– ) < ) Therefore, f ′ ( x ) < when – ∞ < x < – .
Thus, the function f is decreasing in (– ∞ , – ). Consider the interval (– , ), i.e., when – < x < . In this case, we have x – < , x + > and x – < (In particular, observe that for x = , f ′ ( x ) = ( x – ) ( x + ) ( x – ) = (– ) ( ) (– ) = > ) So f ′ ( x ) > when – < x < . Thus, f is increasing in (– , ).
Now consider the interval ( , ), i.e., when < x < . In this case, we have x – > , x + > and x – < . So, f ′ ( x ) < when < x < . Thus, f is decreasing in ( , ).
Finally, consider the interval ( ,