Finding a function $f$ with the minimal $\|f'\|_1$ I was wondering about the following question, which I am sure the answer is known. I couldn't quite find it and I would appreciate if someone could tell me. 
Suppose I have a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that 
$f(-1)=0$, $f(0)=1$, $f(1)=0$, and  $f(x) = 0$ if $|x|>1$. I was wondering what is the minimum possible value of  $\int_{\mathbb{R}} |f'(x)| dx$ and what is $f$ that achieves it? 
Thank you very much!

Sketch: Assuming $f\in C^1,$ we have $\int_{-1}^0f' = f(0)-f(1) = 1.$ Hence $\int_{-1}^0|f'| \ge 1.$ The same applies on $[0,1].$ Thus
$$\int_{\mathbb R}|f'| = \int_{-1}^1|f'| \ge 2.$$
Towards finding the minimum value: Let $f$ be any $C^1$ function satisfying the hypotheses that in addition is increasing on $[-1,0]$ and decreasing on $[0,1].$ 
