ADVANCED EXERCISES

(1) For each of the following functions f(x), find the interval on which:
  (i) f(x) > 0; and (ii) f(x) < 0.  
(Note: for some functions, the answer to (i) or (ii) may be the empty set.)


 (a) f(x) = 1/6(x + 1)3(x - 1)

 (b) f(x) = (x - 1)2 - 4

 (c) f(x) = (x2 - 2x - 3)2

 (d) f(x) = -x2(x - 3)2

              3x
 (e) f(x) = ------
            2x - 4

            x2 - 4
 (f) f(x) = -------
              x2

               18
 (g) f(x) = ---------
            (x - 3)2

              -x
 (h) f(x) = -------
            x2 + 3

 (a) f(x) = 1/6(x + 1)3(x - 1)

     x + 1   - - -        + + +          + + +
     x - 1   - - -        - - -          + + +
           ___________o________________o___________
                     -1                1
     f(x)    + + +        - - -          + + +

     f(x) > 0  on  (-,-1)  (1,)
     f(x) < 0  on  (-1,1)


 (b) f(x) = (x - 1)2 - 4
          = x2 - 2x - 3
          = (x - 3)(x + 1)

     x + 1     - - -           + + +                + + +
     x - 3     - - -           - - -                + + +
            ___________o________________________o___________
                      -1                        3
     f(x)      + + +           - - -                + + +

     f(x) > 0 on (-,-1)  (3,)
     f(x) < 0 on (-1,3)


 (c) f(x) = (x2 - 2x - 3)2

     f(x) > 0 for all x on (-,-1)  (-1,3)  (3,) 
     f(x) = 0 for x = -1 , 3

 (d) f(x) = -x2(x - 3)2

     f(x) < 0 for all x on the interval (-,0)  (0,3)  (3,) 
     f(x) = 0 for x = 0 , 3

              3x
 (e) f(x) = ------
            2x - 4

     3x          - - -          + + +          + + +
     2x - 4      - - -          - - -          + + +
              _____________o________________o____________
                           0                2
     f(x)        + + +          - - -          + + +

     f(x) > 0  on  (-,0)  (2,)
     f(x) < 0  on  (0,2)


            x2 - 4
 (f) f(x) = ------
              x2

            (x - 2)(x + 2)
          = --------------
                  x2

     x - 2    - - -        - - -           - - -           + + + 
     x + 2    - - -        + + +           + + +           + + + 
     x2       + + +        + + +           + + +           + + +
            __________o________________o________________o__________
                     -2                0                2
     f(x)     + + +        - - -            - - -           + + +

     f(x) > 0  on  (-,-2)  (2,)
     f(x) < 0  on  (-2,0)  (0,2)


               18
 (g) f(x) = --------
            (x - 3)2

     f(x) > 0  for all x on the interval (-,3)  (3,)


              -x
 (h) f(x) = ------
            x2 + 3

     Since x2 + 3 is always positive, we can just use -x

     -x > 0  ==>  x < 0

     -x < 0  ==>  x > 0

     f(x) > 0  on  (-,0)
     f(x) < 0  on  (0,)