Consider ODE: $\frac{dy}{dx}=f(y),\: x \in \mathbb{R}$ with $y(0)=y(1)=0$ and $f$ satisfies a Lipschitz function.

Question

Let $y: \mathbb{R}\rightarrow \mathbb{R}$  be differentiable with $y(0)=y(1)=0$ and satisfy the ODE: $$\frac{dy}{dx}=f(y),\quad  x \in \mathbb{R}$$ Suppose  $f:\mathbb{R}\rightarrow \mathbb{R}$ is a Lipschitz condition function. Then

1. $y(x)=0$ if and only if $x\in {0,1}$
2. $y$ is bounded.  
3. $y$ is strictly increasing.
4. $\frac{dy}{dx}$ is unbounded