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anonymous

  • one year ago

Find the absolute extrema of...

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  1. anonymous
    • one year ago
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    \[\sqrt[3]{x}\]

  2. anonymous
    • one year ago
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    I already solved for the derivative and got... \[\frac{ 1 }{ 3 }x^{-\frac{ 2 }{ 3 }}\]

  3. anonymous
    • one year ago
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    I'm just not exactly sure what to do next.

  4. Michele_Laino
    • one year ago
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    by the Weierstrass theorem we need of a closed interval of the real line

  5. anonymous
    • one year ago
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    I can't say that I've learned about the Weierstrass theorem.

  6. anonymous
    • one year ago
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    Is that where you just find all the critical points of the closed interval and determine which is the max?

  7. Michele_Laino
    • one year ago
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    yes!

  8. anonymous
    • one year ago
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    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)?

  9. Michele_Laino
    • one year ago
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    your first derivative is always positive, which means that your function is an increasing function

  10. Michele_Laino
    • one year ago
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    of course at the point \(x=0\) your first derivative is not defined

  11. anonymous
    • one year ago
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    So it would have an asymptote?

  12. Michele_Laino
    • one year ago
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    no, since the asymptotes occurs at point \(x\) such that the function (not the first derivative) becomes an infinity

  13. anonymous
    • one year ago
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    Okay, I think I understand.

  14. Michele_Laino
    • one year ago
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    ok! :)

  15. anonymous
    • one year ago
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    Since my function is an increasing function, is there a way to have specific critical points? (like 1,2,3..)

  16. anonymous
    • one year ago
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    Or would it just be infinity?

  17. Michele_Laino
    • one year ago
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    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 \(f\)

  18. anonymous
    • one year ago
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    Okay. Got it. Thank you! :)

  19. Michele_Laino
    • one year ago
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    :)

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