# Math Help - An Integral Inequality #1

1. ## An Integral Inequality #1

Assume $f \in C^{(1)} ([0,1])$,and $f(0)=0,f(1)=1,$ then

$\int_0^1 |f(x)-f'(x)| dx \geq \frac{1}{e}$

thanks very much

2. Originally Posted by Xingyuan
Assume $f \in C^{(1)} ([0,1])$,and $f(0)=0,f(1)=1,$ then

$\int_0^1 |f(x)-f'(x)| dx \geq \frac{1}{e}$

thanks very much
Use the fact that $\forall x \in [0,1], |e^{-x}(f(x) - f'(x))| \leq |f(x) - f'(x)|$

$\Rightarrow \int^1_0 |e^{-x}(f(x) - f'(x))|\, dx \leq \int^1_0|f(x) - f'(x)|\, dx$

$\Rightarrow \int^1_0 d(|e^{-x}f(x)|) \leq \int^1_0|f(x) - f'(x)|\, dx$

$\Rightarrow |e^{-x}f(x)|\bigg{|}_{0}^{1} \leq \int^1_0|f(x) - f'(x)|\, dx$

$\Rightarrow |e^{-1}f(1)| - |e^{-0}f(0)| \leq \int^1_0|f(x) - f'(x)|\, dx$

$\Rightarrow \int^1_0|f(x) - f'(x)|\, dx \geq \frac1{e}$
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