Exercise 4.  Let  .  

Evaluate the following, where R represents the rectangular region defined by the set  .   

4 (c).  .  

Solution 4 (c).

See text and/or instructor's solution manual.

Answer.   .  

Solution.  Since   we have  

          .  

Solutin part (i).  For   ,  .

This function is increasing on the interval  ,  so the maximum occurs at the right endpoint where  ,  

the maximum of    occurs at  ,  and  .  

                    The maximum of    on the interval      is   .  

Solutin part (ii).  For   ,  .

This function is increasing on the interval  ,  so the maximum occurs at the left endpoint where  ,  

the maximum of    occurs at  ,  and  .  

                    The maximum of    on the interval      is   .  

Solutin part (iii).  For   ,  .

This function is increasing on the interval  ,  so the maximum occurs at the right endpoint where  ,  

the maximum of    occurs at  ,  and  .  

                    The maximum of    on the interval      is   .  

Solutin part (iv).  For   ,  .  

This function is increasing on the interval  ,  so the maximum occurs at the left endpoint where  ,  

the maximum of    occurs at  ,  and  .  

                    The maximum of    on the interval      is   .  

Comparing the above values

                        

We conclude that the global maximum occurs at the point    and  .

          Graph of   over     and   .

This solution is complements of the authors.

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