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Math Help - Strange critical number problem

  1. #1
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    Strange critical number problem

    Find all the critical numbers of y = x (lnx)^2 on the interval (0, infinity)

    Obviously, two of the critical numbers are x = e^(-2) and x = 1

    I also included zero and infinity as two other critical numbers. But my book didn't include them! I understood that infinity isn't considered as a critical number (new info for me) but what about zero??? lnx at zero, not to mention the endpoint of the interval, is undefined and doesn't that make the slope undefined as well?
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  2. #2
    Senior Member mollymcf2009's Avatar
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    Quote Originally Posted by Kaitosan View Post
    Find all the critical numbers of y = x (lnx)^2 on the interval (0, infinity)

    Obviously, two of the critical numbers are x = e^(-2) and x = 1

    I also included zero and infinity as two other critical numbers. But my book didn't include them! I understood that infinity isn't considered as a critical number (new info for me) but what about zero??? lnx at zero, not to mention the endpoint of the interval, is undefined and doesn't that make the slope undefined as well?
    e^{-2} and 1 are your only critical values. f'(x) does not change its sign at zero, so why would you include that as a critical value?
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    But since zero is an endpoint of the function, the slope is therefore undefined at that point. Since the slope is undefined at that point, that must mean zero is a critical number as well.......... or am I missing something?
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  4. #4
    Senior Member mollymcf2009's Avatar
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    Quote Originally Posted by Kaitosan View Post
    But since zero is an endpoint of the function, the slope is therefore undefined at that point. Since the slope is undefined at that point, that must mean zero is a critical number as well.......... or am I missing something?

    According to your domain (0, \infty) zero was not included in the domain.

    If it were included, your domain would be stated as [0,\infty)
    with a bracket.

    If it IS in the domain, or however your problem reads, then yes, it would be a critical number because it is undefined at zero. Maybe I'm being too nitpicky sorry
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    Oh I see! You're completely right, thanks! I understand now.
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