anonymous
  • anonymous
I don't understand how to find the absolute extrema on an open interval.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
on an open interval?
anonymous
  • anonymous
For example, if the domain is on the interval negative infinity to positive infinity. Not a closed interval.
anonymous
  • anonymous
well, that interval is not closed...

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

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