Question

I don't understand how to find the absolute extrema on an open interval.

Answer 1

on an open interval?

Answer 2

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

Answer 3

well, that interval is not closed...

Answer 4

i understand that intuition may tell you that it is

Answer 5

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

Answer 6

okay

Answer 7

that is easier to approach

Answer 8

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

Answer 9

do you see that? or should we discuss it?

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

oh, can you please explain that

Answer 14

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

Answer 15

correct

Answer 16

of degree 4

Answer 17

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

Answer 18

yes, I understand that

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

right

Answer 23

so, find the critical points

Answer 24

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

Answer 25

its at x = 1 max = 1

Answer 26

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

Answer 27

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

Answer 28

yes I mean that too

Answer 29

Thank you so much! Just one last question..

Answer 30

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

Answer 31

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

Answer 32

oh ok

Answer 33

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

Answer 34

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

Answer 35

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

Answer 36

and the critical points are -1 and 1

Answer 37

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

Answer 38

right, so it will have no "absolute" extrema

Answer 39

do you see why?

Answer 40

No I can't seem tosee it

Answer 41

its all in that one word "absolute",

absolute extreme , implies that the function achieves no values greater/less than, respectively, that value

Answer 42

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

Answer 43

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

Answer 44

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

Answer 45

awesome, good luck!