Example of integrable, differentible sequence of functions

Question

Is there a sequence of functions ${f_n(x)}$ satisfying $$\int_{0}^{1} |f_n| dx=1/n,\quad \mbox{and}\quad \int_{0}^{1} |f_n '| dx=1?$$ I was looking for a sequence of functions satisfying $L^1$-norm is decreasing to $0$ and $L^1$-norm of derivative of functions is a positive constant.

Answer 1

You may try with $f(x)=x^{n-1}$

Answer 2

A more "exotic" example. The graph of $f_n\in W^{1,1}(0,1)$ can be a strip of $n$ right isosceles triangles with all the hypotenuses of size $\frac{1}{n}$ along the segment $[0,1]$. Then $$\int_{0}^{1} |f_n| dx=\frac{1}{4n}\to 0, \quad\mbox{and}\quad \int_{0}^{1} |f_n '| dx=1.$$