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Math Help - A question about graphing natural logs

  1. #1
    zep
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    A question about graphing natural logs

    Hi all! I'm new to the forums. I'm finishing up high school calculus and was looking for some help with a question.
    I was asked to find the regions of increase and decrease for y=xln(x^2)

    First, I found the derivative to be 2(1+lnx)
    Next, to find when the function is increasing, I had to solve for lnx>-1
    Then I went x>e^-1, so x>1/e
    This suggests to me that the function is increasing beyond x=1/e. While this is true, when I graphed the initial function, it appears as if the graph changes directions twice, whereas I only found it to change direction once.

    Could someone please indicate where my error is? Thank you!
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  2. #2
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    Re: A question about graphing natural logs

    Quote Originally Posted by zep View Post
    Hi all! I'm new to the forums. I'm finishing up high school calculus and was looking for some help with a question.
    I was asked to find the regions of increase and decrease for y=xln(x^2)

    First, I found the derivative to be 2(1+lnx)
    Next, to find when the function is increasing, I had to solve for lnx>-1
    Then I went x>e^-1, so x>1/e
    This suggests to me that the function is increasing beyond x=1/e. While this is true, when I graphed the initial function, it appears as if the graph changes directions twice, whereas I only found it to change direction once.

    Could someone please indicate where my error is? Thank you!
    $y(x)=x \ln(x^2)$

    $y^\prime(x)=2+\ln(x^2)$

    $y^\prime(x)=0 \Rightarrow x=-e^{-1} \vee x=e^{-1}$

    You'll find $y^\prime(x)$ decreases on $(-\infty, 0)$ and increases on $(0, \infty)$

    basically you forgot about the negative real axis which, because $x^2$ is the argument to $\ln()$, is included in the domain of $y(x)$
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