Calculus-Sec-4-2-Solution

4.2   The Shape of a Graph            by SGLee - HSKim - JHLee

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. .

(a) local maximum:

local minimum:

(b) increasing on

decreasing on

(c) inflection point at

concave down on

concave up on

2.

(a) local minimum:

(b) increasing on

decreasing on

(c) inflection point at

concave down on

concave up on

3. .

(a) local maximum:

local minimum:

(b) increasing on

decreasing on

(c) inflection point at

concave down on

concave up on

To find local maximum and minimum, use the next sage code.

To find inflection points, we calculate the second derivative.

4.

(a) local maximum :  No

local minimum :

(b) increasing on

decreasing on

(c) inflection point : No

concave up on

5.

(a) Maximum : 1, Minimum : 0

(b) interval of increase :

interval of decrease :

(c) inflection point at  변곡점은 0.722, -0.722

concave down on

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

6.

(a) inflection point at

(b) concave down on

concave up 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.

(a) critical point at ,  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 points :  and

(b) concave down on

concave up on

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

12.

vertical asymptote

horizontal asymptote

13.

vertical asymptote

horizontal asymptote

14.

horizontal asymptote

15.

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.

and

and ;     and

(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.

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

and .

Note that  and . Hence  has local maximum at  and local minimum at .  Therefore, if  there will be  roots.

20. Find the minimum constant  for 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. Use CAS to find  and  when .

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

local maximum :

local minimum : No

inflection point :

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

local maximum :

local minimum : No

inflection point :