How do I find and classify the critical points of $x^2 \sin (\frac 1x)$

Find the non-zero critical points of $f$ and the signs of the values of $f$ at these non zero critical points.

Determine the behaviour of $f$ between the non zero critical points and the types of non-zero critical points

I understand that this function is oscillating in between $(-0.3183, 0)$ and $(0.3183,0)$ and have also proven that $f'(0) = 0$ using the formal derivative definition as well as the squeeze theorem.

I also found $f'(x) = 2x\sin \frac 1x - \cos \frac 1x$ but am having difficulty finding a way to list and classify the critical points in a general manner.

Any help is appreciated!

• Nothing like trying to describe infinitely many critical points on a finite interval, what form of solution are you hoping for. – David Jun 8 '16 at 3:35
• Haha sure sounds like a bit of a feat, I'm honestly not too sure myself. After reading the question, I assumed there would be some sort of general solution to express all the critical points (or perhaps rearranging the subject for x in the derivative?). But yeh, still trying to get my head around analysis and approaching questions. – Astro551 Jun 8 '16 at 3:56
• Interesting, I'll give it a thought. To me, stating all solutions to $xtan(1/x) = 1/2$ would be a satisfactory description of the set, but maybe there is something better we can do... – David Jun 8 '16 at 3:58