I'm trying to find the minimum and maximum points of the following equation: $y = x \sin(\ln|x|)$ where $x > 0$ which, when graphed, looks something like this:
I tried deriving the equation using the chain rule:
$\begin{align}\frac{dy}{dx} &= \frac{d}{dx}\left(x \cdot \sin(\ln|x|)) \right) \\ & = \sin(\ln|x|) + x \cos(\ln|x|) \frac{1}{x} \\ & = \sin(\ln|x|) + \cos(\ln|x|) \end{align}$
...and setting it equal to zero:
$0 = \sin(\ln|x|) + \cos(\ln|x|)$
I set $\ln|x|$ equal to $\frac{3\pi}{4}$ or $\frac{7\pi}{4}$ (because using the unit circle, using those two should cause the sin and cos to cancel out), so that $x = e^{{3\pi}/{4}}$ or $x = e^{{7\pi}/{4}}$, but both of those are clearly far more higher then the x-coordinates of the min and max points in the graph.
How can I find the min and max x-coordinates of this equation? This is a homework problem I've been puzzling over for some time, so either hints or full answers would be appreciated.