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

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

Mathematics
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on an open interval?
For example, if the domain is on the interval negative infinity to positive infinity. Not a closed interval.
well, that interval is not closed...

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i understand that intuition may tell you that it is
f(x) = 4x^3- 3x^4 on the interval negative infintiy to positive infinity
okay
that is easier to approach
first note that \[\lim_{x\rightarrow \infty}f(x)=\lim_{x\rightarrow- \infty}f(x)=-\infty\]
do you see that? or should we discuss it?
My prof. taught us to find the derivative and at the critical points, weshould look how thederivative behaves, but that doesn't always work..
yes, but we are doing some preliminary investigation about the behavior of this function
my point is, this function will have an absolute max because of the result of those limits
oh, can you please explain that
the give f(x) is a polynomial, yes?
correct
of degree 4
so its end behavior will be the same at \[\pm\infty\]
yes, I understand that
in this case since the leading coefficient is (-1), both ends go to \[-\infty\]
okay so whatever this function does, it can not have an "absolute" minimum right?
no, it's suppose to only have an absolute max
right
so, find the critical points
i.e. where \[\frac{df}{dx}=0\]
its at x = 1 max = 1
So, I understood we always look at the leading coefficient of the polynomial?
yes, it as well as the degree will tell you a lot
yes I mean that too
Thank you so much! Just one last question..
what level of calculus are you studying and what are your future plans in math?
this is calc 1 and the onlysemester of calc i have to take.. thankfully! I'm a pharmacy major..
oh ok
good luck then, i have had many students that were pre-pharm
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
so the behavior of function: from negative infinty to positive infinity..
and the critical points are -1 and 1
but the solution says there is no extrema and I don'tunderstand why
right, so it will have no "absolute" extrema
do you see why?
No I can't seem tosee it
its all in that one word "absolute", absolute extreme , implies that the function achieves no values greater/less than, respectively, that value
if your function runs off to \[+\infty\] one way and \[-\infty\] the other, then there is no absolute max/min
There are however possible "local" max/min which is where many students get confused
Oh okay, I will look into it more... Thank you so much foryour help! I really appreciate it!
awesome, good luck!

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