4.1 Extreme values of a function

1-4. Determine if the following statement is true or false. Explain your answer.

1. If is a continuous function, then has a maximum at only one point in .

 False. Consider

2. One can apply the Mean Value Theorem to

   on .

 False

3. If a continuous function has a extreme value on , then has a absolute maximum or minimum value on .

 False

4. For a continuous function , has only one zero provided is strictly decreasing.

 True

5-8. Find all critical numbers of given functions.

5. .

http://matrix.skku.ac.kr/cal-lab/cal-4-1-5.html

[x == 2, x == (3/2), x == (5/3)]

6. .

7. .

http://matrix.skku.ac.kr/cal-lab/cal-4-1-7.html  

[x == 1, x == -1]

8. .

http://matrix.skku.ac.kr/cal-lab/cal-4-1-8.html

[x == -1, e^x == 0]

9-13. Find all local maximum and minimum if it exists.

9. .

http://matrix.skku.ac.kr/cal-lab/cal-4-1-9.html

 No local minimum and maximum

10. .

http://matrix.skku.ac.kr/cal-lab/cal-4-1-10.html

 No local maximum, local minimum

11. .

http://matrix.skku.ac.kr/cal-lab/cal-4-1-11.html

sin(x)*cos(x)/abs(sin(x)) 

[x == 0, x == 1/2*pi]

12. .

      if , , then has a local minimum at .

      if , , then has a local minimum at .

      if , , then has a local maximum at .

13. , .

       then has a local maximum at

14-17. Find the intervals where the function is increasing or decreasing.

14. .

http://matrix.skku.ac.kr/cal-lab/cal-4-1-14.html

 Increasing on ,  decreasing on

15. .

http://matrix.skku.ac.kr/cal-lab/cal-4-1-15.html

 Increasing on , decreasing on

[x == 1]

16. .

      Increasing on , decreasing on

17-19. Prove the inequality.

17. .

      So, for

18. , .

      So, (see Exercise 21).

19. .

      So, (see Exercise 21).

20. Let . Find the number of zeros and give a proper interval containing all zeros.

http://matrix.skku.ac.kr/cal-lab/cal-4-1-20.html 

 7 zeros in

21. Suppose and for all . Using the Mean Value Theorem show for all .

 By the Mean Value Theorem,

      .

22. Show that for .

 By Excercise (21),

      Then, .

23-24. Prove the inequality using the Mean Value Theorem.

23. .

Let , then by the Mean Value Theorem there exists in such that

      . Since,

      , we have

      .

24. .

Let , than there exists in such that

      . Since,

      , we have

 .

25. Prove that the inequality for all and using the Mean Value Theorem.

Let , , than there exists in such that

      Thus,.

26. Let be a continuous function. Prove that the only functions satisfying for all are of the form , by the Mean Value Theorem.

 Applying Mean Value Theorem to on , there exists in such that

       for all . This implies

      Now set , , then for all .

27. Suppose is continuous on and differentiable on . Prove that there exists such that .

Let , then

      .

      By Rolle's theorem,

      there exist such that , so

      .

28. Prove Fermat's Theorem.(Give either reference or statement)

 Suppose, for the sake of definiteness, that has a local maximum at . Then, if is sufficiently close to . This implies that if is sufficiently close to 0, with being positive or negative, then and therefore (1). We can divide both sides of this inequality by a positive number. Thus, if and is sufficiently small, we have

      .

      Taking the right-hand limit of both sides of this inequality, we get

      But since exists, we have

      and so we have shown that .

      If , then the direction of the inequality (1) is reversed when we divide by :

      .

      So, taking the left-hand limit, we have

      We have shown that and also that . Since both of these inequalities must be true, the only possibility is that

29. Prove Rolle's Theorem.(Give either reference or statement)

Since , if has local extremum at any one of the end point, the function is constant.

      Hence the result is clearly.

      Suppose that the maximum is obtained at an interior point of . We shall examine the above right- and left-hand limits separately.

      For a real such that is in , the value is smaller or equal to because attains its maximum at . Therefore, for every ,

      ,

      hence

      ,

      where the limit exists by assumption (it may be minus infinity).

      Similarly, for every , the inequality is reversed because the denominator is now negative and we get

      ,

      hence

      ,

      where the limit might be plus infinity.

      Finally, when the above right- and left-hand limits agree, then the derivative of at must be zero.

30. Prove Increasing and Decreasing Test.(Give either reference or statement)

 (a) Let and be any two numbers in the interval with . According to the definition of an increasing function we have to show that .

          Because we are given that , we know that is differentiable on . So, by the Mean Value Theorem, there is a number between and such that

          . (1)

          Now by assumption and because . Thus the right side of (1) is positive, and so

           or .

          This shows that is increasing.

          Part (b) is proved similarly.

31. Prove the cases (b), (c) of the First Derivative Test.(Give either reference or statement)

 (b) Let us choose at sufficiently near such that . By the Mean Value Theorem, there exist with and such that

           and

          .

          Since for and for , is decreasing on and increasing on . Hence has a local minimum at .

       (c) Let us choose at sufficiently near such that . By the Mean Value Theorem, there exist with and such that

           and

          .

          Since for and for . Hence has neither local maximum nor local minimum at .

32. Prove the case (b) of the Second Derivative Test.

 (b) Suppose we have . Then

          Thus, for sufficiently small we get

          which means that if so that is decreasing to the left of , and that if so that is increasing to the right of . Now, by the first derivative test we know that has a local maximum at .