Let $V\subset \subset U\subset \mathbb{R}^1$. Can we construct a Lipschitz cut-off function $f$ such that $$ f=0 \quad \text{on} \quad V ,\quad f=-1 \quad \text{on} \quad \mathbb{R}^1 \setminus U,\quad f^2-|f'|\geq -C \quad \text{a.e.}\quad \text{on} \quad U\setminus V $$ hold？
Is there a concrete example of such a function?
Could you please give me some help with details? Thanks in advance.