$f$ maximum in $0$, $[0,\epsilon]$ not decreasing infinitely close to $0$

Question

Problem:

Construct a two times differentiable function $f: (-1,1)\to \mathbb{R}$ with the following characteristics:

1. $f$ has a strict local maximum in $0$
2. There is no $\epsilon>0$, such that $f$ is decreasing in $[0,\epsilon]$
3. There is no $\epsilon>0$, such that $f$ is increasing in $[-\epsilon,0]$

Source: Real analysis coursebook from Otto Forster, chapter about extremas, l'hopital, convexity

So far I tried: I had an idea with functions, that oscilate infinitely when they tend to $0$, such as $f(x) =\sin{\frac{1}{x^2}}$, but at $0$ they would have some value that is defined, such that it would all be differentiable. I failed at finding such a value, that it would be differentiable everywehere

Answer 1

A possibility is$$f(x)=\begin{cases}-x^4\left(\cos\left(\frac1x\right)+2\right)&\text{ if }x\ne0\\0&\text{ otherwise.}\end{cases}$$Its graph near $0$ can be seen in the picture below: