## anonymous 5 years ago I don't understand how to find the absolute extrema on an open interval.

1. anonymous

on an open interval?

2. anonymous

For example, if the domain is on the interval negative infinity to positive infinity. Not a closed interval.

3. anonymous

well, that interval is not closed...

4. anonymous

i understand that intuition may tell you that it is

5. anonymous

f(x) = 4x^3- 3x^4 on the interval negative infintiy to positive infinity

6. anonymous

okay

7. anonymous

that is easier to approach

8. anonymous

first note that $\lim_{x\rightarrow \infty}f(x)=\lim_{x\rightarrow- \infty}f(x)=-\infty$

9. anonymous

do you see that? or should we discuss it?

10. anonymous

My prof. taught us to find the derivative and at the critical points, weshould look how thederivative behaves, but that doesn't always work..

11. anonymous

yes, but we are doing some preliminary investigation about the behavior of this function

12. anonymous

my point is, this function will have an absolute max because of the result of those limits

13. anonymous

oh, can you please explain that

14. anonymous

the give f(x) is a polynomial, yes?

15. anonymous

correct

16. anonymous

of degree 4

17. anonymous

so its end behavior will be the same at $\pm\infty$

18. anonymous

yes, I understand that

19. anonymous

in this case since the leading coefficient is (-1), both ends go to $-\infty$

20. anonymous

okay so whatever this function does, it can not have an "absolute" minimum right?

21. anonymous

no, it's suppose to only have an absolute max

22. anonymous

right

23. anonymous

so, find the critical points

24. anonymous

i.e. where $\frac{df}{dx}=0$

25. anonymous

its at x = 1 max = 1

26. anonymous

So, I understood we always look at the leading coefficient of the polynomial?

27. anonymous

yes, it as well as the degree will tell you a lot

28. anonymous

yes I mean that too

29. anonymous

Thank you so much! Just one last question..

30. anonymous

what level of calculus are you studying and what are your future plans in math?

31. anonymous

this is calc 1 and the onlysemester of calc i have to take.. thankfully! I'm a pharmacy major..

32. anonymous

oh ok

33. anonymous

good luck then, i have had many students that were pre-pharm

34. anonymous

Thank you. One last question, if the equation was f(x) = 2x^3 - 6x +2.. the polynomial is of degree 3 and leading coef is positive

35. anonymous

so the behavior of function: from negative infinty to positive infinity..

36. anonymous

and the critical points are -1 and 1

37. anonymous

but the solution says there is no extrema and I don'tunderstand why

38. anonymous

right, so it will have no "absolute" extrema

39. anonymous

do you see why?

40. anonymous

No I can't seem tosee it

41. anonymous

its all in that one word "absolute", absolute extreme , implies that the function achieves no values greater/less than, respectively, that value

42. anonymous

if your function runs off to $+\infty$ one way and $-\infty$ the other, then there is no absolute max/min

43. anonymous

There are however possible "local" max/min which is where many students get confused

44. anonymous

Oh okay, I will look into it more... Thank you so much foryour help! I really appreciate it!

45. anonymous

awesome, good luck!