1. ## 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!

2. ## Re: A question about graphing natural logs

Originally Posted by zep
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 $y(x)$