4.2 The Shape of a Graph

1-5. Find the local maximum and minimum values of . In addition, find the intervals on which is increasing and decreasing, and the intervals of concavity and the inflection points, sketch a graph of .

1. .

http://matrix.skku.ac.kr/cal-lab/cal-4-2-1.html

      (a) local maximum:  

         local minimum:

      (b) increasing on

         decreasing on

      (c) inflection point at

         concave down on

         concave up on

2. .

      http://matrix.skku.ac.kr/cal-lab/cal-4-2-2.html

      (a) local minimum:

      (b) increasing on

         decreasing on

      (c) inflection point at

         Concave down on

         Concave up on

4*x^3-68*x

[x == -sqrt(17), x == sqrt(17), x == 0]

12*x^2-68

[x == -1/3*sqrt(3)*sqrt(17), x == 1/3*sqrt(3)*sqrt(17)]

3. .

      (a) local maximum:

         local minimum:

      (b) increasing on

         decreasing on

      (c) inflection point at

         Concave down on

         Concave up on

4. .

 (a) local maximum :  No

         local minimum :

      (b) increasing on

         decreasing on

      (c) inflection point : No

         Concave up on

5. .

http://matrix.skku.ac.kr/cal-lab/cal-4-2-5.html

 (a) Maximum : 1, Minimum : 0

      (b) interval of increase :

          interval of decrease :

      (c) inflection point at

          Concave down on

-2*(tanh(x^2)^2 - 1)*x

6-11. Find the inflection points of In addition, find where the graph of is concave upward or concave downward.

6. .

      (a) inflection point at

      (b) concave up on

         concave down on

7. .

      (a) inflection point at

      (b) concave up on

         concave down on

8. , .

      (a) inflection point at

      (b) concave down on

         concave up on

9.

http://matrix.skku.ac.kr/cal-lab/cal-4-2-9.html

[x == (1/3)]

      (a) inflection point at

      (b) concave down on

         concave up on

10. .

      (a) inflection point at

      (b) concave up on

         concave down on

11.

      (a) inflection point :

      (b) concave up on

         concave down on

12-15. Find the vertical and horizontal asymptotes of .

12.

      vertical asymptote

      horizontal asymptote

13.

http://matrix.skku.ac.kr/cal-lab/cal-4-2-13.html

 vertical asymptote

      horizontal asymptote

14. .

      horizontal asymptote

15.

http://matrix.skku.ac.kr/cal-lab/cal-4-2-15.html

 vertical asymptote :

horizontal asymptote : No

16-18. Sketch the graph of using the following information.

       (a) Find the local maximum and minimum values of .

       (b) Find the intervals of increase or decrease.

       (c) Find the inflection points of and intervals of concavity.

       (d) Find the vertical and horizontal asymptotes.

16.

 (a) local maximum : No

         local minimum : No

      (b) decreasing on

      (c) inflection point at

         concave up on

         concave down on

      (d) vertical asymptote :

         horizontal asymptote :

17.

      ,

      (a) local minimum at

      (b) increasing on

         decreasing on

      (c) inflection point : No

         concave up on

      (d) vertical asymptote:

        horizontal asymptote: No

18.  , .

      ,

      (a) local minimum at

         local maximum :

      (b) increasing on

         decreasing on

      (c) inflection point at

         concave up on

         concave down on

      (d) vertical asymptote: No

          horizontal asymptote:

19. Find all the values of a such has different roots.

  will have one local maximum and one local minimum. There will be roots if such and only if the maximum is positive and the minimum is negative.

      Therefore, if there will be roots.

20. Find the minimum constant on which for all real .

      ,

      local minimum at

      Therefore,

21. Find so that has two

    inflection points at and .

       and so

      ,

       and so

22. Let .

      (a) Find and .

      (b) Find the vertical and horizontal asymptotes of .

      (c) Sketch the graph of using (a) and (b).

      (a)

      (b) ,

         vertical asymptote

         horizontal asymptote

      (c)

23. Let and be increasing functions. Prove that is increasing function.

 For , ,

      Therefore, is increasing.

24. Prove the Concavity Test.

 (a) By Increasing Test

      If on an interval then is increasing on .

      So is concave upward on .

      Part (b) is proved similarly.

25-27. Use CAS.

25. Find and when .

http://matrix.skku.ac.kr/cal-lab/cal-4-2-25.html

[df == 4*x^(1/3) + 2/x^(2/3), dff == 4/3/x^(2/3) - 4/3/x^(5/3)]

26. Let . Find the local maximum, minimum values and inflection points of . Sketch the graph of .

http://matrix.skku.ac.kr/cal-lab/cal-4-2-26.html

 local maximum :

      local minimum : No

      inflection point :

27. Let . Find the local maximum, minimum values and inflection points of .

http://matrix.skku.ac.kr/cal-lab/cal-4-2-27.html

 local maximum :

      local minimum : No

      inflection point :