On a class of oscillating integrals

The following result is proved in the paper: if for some real $A>0$ and some natural number $n>1$ for all $x$ from $[0,1]$ we have the inequality $|f^{(n)}(x)|\geq A,$ then the following estimate is valid:
$$|I|=\left|\int_0^1\limits\rho(f(x))~dx\right|\leq\min{\{1;4nA^{-1/n}\}},$$
where $\rho(t)=0,5-\{t\}.$