Is the function $y=x/2 + x^{2}\sin(1/x)$ monotonic near $0$?
The derivative $f'$ obviously goes positive and negative near $0$, because $$f'(x)= \frac12 + 2x\sin(1/x) - \cos(1/x))$$ Does that mean that $f$ is not monotonic near $0$?
Is the function $y=x/2 + x^{2}\sin(1/x)$ monotonic near $0$?
The derivative $f'$ obviously goes positive and negative near $0$, because $$f'(x)= \frac12 + 2x\sin(1/x) - \cos(1/x))$$ Does that mean that $f$ is not monotonic near $0$?