SKKU-Calculus-Sec-13-8 Extrema of Multivariate Functions-New

   13.8     Extrema of Multivariate Functions          by SGLee - HSKim- SWSun-JHLee, 오교혁




1. Let . Find the critical points of  and classify them.    



      Solve  and .

      So we have critical points,  or . If , then .

      If , then we have .

      The critical points are .

      Next, we consider second order partial derivatives to get .

      Then  and

       thus we obtain   at points .

      This implies that  are saddle points.

      At points , we observe that   .

      Moreover, since 

       has a local maximum  at .

      On the other hand, due to 

       has a local       minimum  at .


위에서 구한 를 saddle point, local maximum, local minimum으로 각각 분류하자.

즉, 은 saddle point이고, 은 local maximum이다.

2. Find the extreme values of the function  when .







       : critical points

      =>  has no local minimum. or maximum at .

     At ,


       , so  has a local maximum  at .


3-4. Locate the maxima, minima, and saddle points of the functions.









      Try this on your own.

5. Let . Answer the following:

   (a) Find points of local maximum/minimum and a saddle point when .

   (b) Give a condition on  for the case when  has only one critical point.



          =>  (1,1) is a saddle point.

          =>  are points of local minimum..






       If  has only one critical point  has a solution and 

        should not have a solution. So .


6. Find maximum value of  on .




      So the critical point is  and thus critical value is 

      Let ,

       and .

      On , we have  and  ,


      On , we have  and  ,


      On , we have  and  ,


      On , we have  and  .


      So the maximum value is 2.  

7. Find the absolute maximum and minimum of   in the domain  

    which is a closed triangle made of three points (0. 0), (2, 1), (1, 2). 



            => critical point : ,  


   (2)  1.  moves on 

                => The absolute maximum , and the absolute minimum  on .

            2.  moves on 

                => The absolute maximum , and the absolute minimum  on .

            3.  moves on 

                => The absolute maximum   , and the absolute minimum  on .

            Hence the absolute maximum is 2 and the absolute minimum is 0.

8. Find the absolute maximum and minimum values   on the disk  D:



       interior of  :


      Then  implies 

       If  implies 

      Thus, we get the critical points 

       If  then  This implies .

       Critical points are 


       Consider , boundary of  :


       Moreover,  is smallest when  and largest when   But  

       Thus on D the absolute maximum of  is  and the absolute minimum is 

9. Find the Taylor series for the function  at the point .










10. Expand the Maclaurin series for the function .









      In general,




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                                                    Back to Part II