anonymous
  • anonymous
Find the absolute extrema of...
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
\[\sqrt[3]{x}\]
anonymous
  • anonymous
I already solved for the derivative and got... \[\frac{ 1 }{ 3 }x^{-\frac{ 2 }{ 3 }}\]
anonymous
  • anonymous
I'm just not exactly sure what to do next.

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Michele_Laino
  • Michele_Laino
by the Weierstrass theorem we need of a closed interval of the real line
anonymous
  • anonymous
I can't say that I've learned about the Weierstrass theorem.
anonymous
  • anonymous
Is that where you just find all the critical points of the closed interval and determine which is the max?
Michele_Laino
  • Michele_Laino
yes!
anonymous
  • anonymous
Awesome! One question though, how would I find the critical points of my derivative since I can't really isolate x (at least from what I can see)?
Michele_Laino
  • Michele_Laino
your first derivative is always positive, which means that your function is an increasing function
Michele_Laino
  • Michele_Laino
of course at the point \(x=0\) your first derivative is not defined
anonymous
  • anonymous
So it would have an asymptote?
Michele_Laino
  • Michele_Laino
no, since the asymptotes occurs at point \(x\) such that the function (not the first derivative) becomes an infinity
anonymous
  • anonymous
Okay, I think I understand.
Michele_Laino
  • Michele_Laino
ok! :)
anonymous
  • anonymous
Since my function is an increasing function, is there a way to have specific critical points? (like 1,2,3..)
anonymous
  • anonymous
Or would it just be infinity?
Michele_Laino
  • Michele_Laino
as I said before, we need of a closed interval. For example if we have this interval: \([2,3]\) then we have to evaluate \(f(2),f(3)\). We will conclude that \(f(2)\) is the point of minimum, and \(f(3)\) is the point of maximum for function \(f\)
anonymous
  • anonymous
Okay. Got it. Thank you! :)
Michele_Laino
  • Michele_Laino
:)

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