Consider $f(x) = x^2 \sin(x^3)$. Denote $M$ the set of statinary points of this function (points where $f'(x) = 0$). Prove that $M$ consists only of isolated points.
The statement seems obvious for small $x$ but for very large $x$ from the graph of this function it seems that such points start placing extremely close to each other. For a rigorous proof of that statement, I think I should prove that if $x_1$ is such that $2x_1 \sin (x_1^3) + 3x_1^4 \cos (x_1^3) = 0$ then $\exists \ \epsilon > 0 \ \ \forall \ x: |x - x_1|< \epsilon \ \to 2(x_1+x) \sin((x_1 + x)^3) + 3 (x_1 + x)^4 \cos ((x_1 + x)^3) \neq 0$. But I don't know, how to show it.
Could you please give me any hints? Thanks in advance!
Just a short hint:
Examine the second derivative. It should be nonzero at a critical point, hence your desired statement follows.