Problem 692 (difficulty: 4/10)

Let

\(\displaystyle f(x)=\begin{cases} x+2x^2 \cdot \sin\frac1x , & \text{if $x \ne 0 ,$} \cr 0, & \text{if $x = 0.$}\cr \end{cases} \)

Show that \(\displaystyle f'(0)>1,\) but \(\displaystyle f\) is not monotone increasing in any neighborhood of \(\displaystyle 0\).

Give me another random problem!