prove that $\int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}|f'(x)|dx$ I would appreciate if somebody could help me with the following problem
Q: prove that 
$$\int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}|f'(x)|dx$$
where 
  $f'(x)$ is continuous on $(0,1)$, and $f(0)=0$.

Using the fundamental theorem of  calculus and $f(0)=0$ we obtain $f(x)=\int_0^x f'(t)dt$ and hence $$\int_0^1 |f(x)|dx=\int_0^1 \left|\int_0^x f'(t)dt\right|dx \leq \int_0^1 \int_0^x |f'(t)|dt\ dx\leq \int_0^1 \int_0^1 |f'(t)|dt\ dx = \int_0^1 |f'(t)|dt$$ since $|f'(t)|\geq 0$.
