## anonymous 5 years ago need help finding intervals of increse....max and min...intervals of concavity up and down...and point of inflection f '(x)=3e^x/(3+e^x)^2

1. amistre64

so this is our y' right?

2. amistre64

e^x never equals zero, so there are no critical points

3. anonymous

ok but i get an interval of increase right from some number to infinity

4. amistre64

I would say yes, but to check for concavity we tend to use the 2nd derivative right?

5. anonymous

well i know for sure that my function is increasing so there has to be interval of increasing but i dont know the starting point

6. amistre64

e^x is never a negative number, so all of its positive right?

7. anonymous

figured it out it from -infinity to infinity for increase variables ok now for concavity...

8. amistre64

concave up is my guess.... but use 2nd derivative to be certain

9. anonymous

i meant increase interval is -infinity to infinity

10. amistre64

(3e^x) (3+e^x)^-2 -6 e^2x (3e^x) --------- + ----------- right? (3+e^x)^3 (3+e^x)^2

11. amistre64

3 e^2x --------- (3+e^x)^3

12. amistre64

its always concave up....

13. anonymous

my sources say otherwise and i am looking for the intervals from which it is concave up and down

14. amistre64

i mighta goofed on the 2nd integral.... better double check

15. anonymous

$f"(x)=-(3e^{2x}-9e^x)/e ^{3x}+9e ^{2x}+27e^x+26$

16. amistre64

3e^x( e^3x -2e^2x +9e^x ) ------------------------ (3+e^x)^4

17. anonymous

no it not to the fourth power...

18. amistre64

havent expanded the bottom to see if anything cancels, but thats just a left over from the quotient rule

19. amistre64

it either that, or my heads got all jumbled..... which is very likely :)

20. anonymous

oh no i think things cancel i am just saying the function i have is the correct function already checked it ....lol

21. amistre64

my top gets to be: -3e^x (e^x -1) i cant get it to match yours

22. amistre64

-3e^x (e^x -3)

23. amistre64

now it does lol

24. anonymous

yea there you go so x=ln 3

25. amistre64

yeah, when e^x = 3 x = ln(3) thats right... we get an inflection right?

26. amistre64

everything to the left of that would be (+) and everything to the right of it would be(-)

27. anonymous

yea...problem one complete now for two....consider the function f(x)=2x+4/3x+2 for this function there are two important intervals (-infinity, A) and (A, infinity) where the function is not defined at A. Find A

28. amistre64

leftside is concave up, and rightside of it os concave down

29. amistre64

A = -2/3 there is a vertical asymptote there and possess no real value.

30. amistre64

nothing calcels top to bottom, so it aint a hole; its just the place where the denominator goes to zero... -2/3

31. anonymous

ok yea that was correct....an i see you clarification

32. anonymous

problem 3 sin^2(x/5) defined at interval [-14.907963,3.226991]

33. anonymous

find where f(x) is concave down, global mini of the function, a local max which is not a global max, the function is increasing on a region....note: some answers must be given in interval notation

34. amistre64

the normal period for pi is about 6 and this stretches it past 6 and goes to 10pi for the period. 10pi is about 31.4

35. amistre64

the amplitude is exaggerated, but thats just the y value....

36. amistre64

sin normally has a max at pi/2 and a min at 3pi/2 so, this one will be at 5pi/2 and 15pi/2 if I am picturing it correctly

37. amistre64

but lets work it :)

38. amistre64

Dx (3 sin^2(x/5)) y' = (3/5) sin(x/5)cos(x/5)

39. amistre64

if sin or cos are zero, we get zeros...

40. anonymous

the fx is sin^2(x/5) sorry its problem 3....

41. amistre64

sin = 0 at 0 and 180, cos = 0 at 90 and 270

42. amistre64

replace (3/5) with (1/5) same setup tho

43. amistre64

but because the normal period is elongated to 10pi, we gotta find those numbers :)

44. amistre64

they are..... 0, 5pi, 5pi/2, 15pi/2

45. amistre64

we can go backwards to... -5pi, -5pi/2, -15pi/2, -10pi

46. amistre64

which of these numbers is in our interval?

47. amistre64

5pi is bigger than 15

48. amistre64

[-14.907963 , 3.226991] -15/2 = -7.5, so -5pi/2 is in our interval 0 is in our interval we got these 2 numbers (-5pi/2) and (0)

49. anonymous

are you using $\sin ^2(x/5)$

50. amistre64

yes, I am using the derivative of that and checking for zeros :)

51. amistre64

the 3 was superfluous and could be tossed on a whim...

52. anonymous

ok

53. anonymous

ok the global mini is at 0 but now we need a local max

54. amistre64

-5pi/2 would be the local max...

55. anonymous

nah it not...i checked

56. amistre64

this is what we look like I think

57. anonymous

our local max was 3.22 from the interval

58. anonymous

i need the interval where it is concave down and region where it is increasing

59. amistre64

computer froze