Exercise 13.  Find specific values of    so that   .  

Solution 13.

See text and/or instructor's solution manual.

Answer.  There are many possibilities, such as   and  .  Explain.

Solution.   

There are many possible counter examples.  For the choice   and    we have

                     ,  

                     ,  

                     .  

Hence,  ,  

and here we have  .  

We are done.   

Aside.  We can let Mathematica check our work.

We are really done.   

Another solution.  There are many possible counter examples.  For the choice   and    we have

                    ,  

                    ,  

                    .  

Hence,  ,  

and here we have  .  

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell