Is there any simple way to find the extreme points to the equation $f(x)= x^2(\sin(1/x)+\cos(1/x))$when $x \in [-1,0) \cup (0,1]$ and $0$ when $x = 0$

Question

Hey I have found the derivative to this equation: $$2x\cdot\sin\left(\frac{1}{x}\right)+2x\cdot\cos\left(\frac{1}{x}\right)+\sin\left(\frac{1}{x}\right)-\cos\left(\frac{1}{x}\right)$$

The things is that i can already see that the endpoint of the interval will end up as the biggest value of the all $x$-values. I want and need to prove that here is no other extreme points inside the interval that is bigger. The only problem is that there are multiple and will take a lot of time prove. Is there a simpler way to do this?