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Let $f(x) = x + 2x^2\sin(1/x)$, with $x\ne0$, and $f(x)= 0$ when $x= 0$. Detrmine if $f(x)$ is increasing or decreasing at $x= 0$

my attempt: using first principle it is easy to see that $f'(0)=1$ however if we find derivative of $f(x)$ as $f'(x) = 1+4x\sin(1/x) -2\cos(1/x)$ ,as $f'(1/(2k\pi)) = -1$ for integral $k$ hence there is no interval around $0$ where $f(x)$ is increasing. So it shouldn't make sense that $f(x)$ is increasing at $x=0$. so how should we handle this? Any help is greatly appreciated.

Andrei
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Ginger bread
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1 Answers1

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$$f(0)=0$$ and

$$f(\epsilon)=\epsilon+2\epsilon^2\sin\frac1{\epsilon}=\epsilon\left(1+2\epsilon\sin\frac1\epsilon\right)>f(0)$$ for all $\epsilon<\frac12$. Similarly $f(-\epsilon)<f(0)$ and the function is indeed growing at $x=0$.