Math Help - Cauchy-Schwarz inequality

1. Cauchy-Schwarz inequality

Hi, I am stuck on this question.

Prove

$|f(L)-f(0)|^2 \le L \int_0^L |f'(x)|^2 dx$ for any function $f \in C^1([0,L])$.

So I know I should use the Cauchy-Schwarz inequality applied to the functions f′ and 1, but I am stuck on how to do it.
Thanks for any help people.

2. Re: Cauchy-Schwarz inequality

$|f(L)-f(0)|=|\int_0^L f'(x)dx|\leq \int_0^L |f(x)|\cdot 1 dx\leq (\int_0^L|f'(x)|^2dx)^{1/2}\cdot (\int_0^L 1^2dx)^{1/2}$

3. Re: Cauchy-Schwarz inequality

Thanks for that, but how do I go on to prove my inequality? I don't see how squaring brings the L in.

4. Re: Cauchy-Schwarz inequality

Originally Posted by iceman954
Thanks for that, but how do I go on to prove my inequality? I don't see how squaring brings the L in.
What is

$\int_{0}^{L}1^{2}\,dx?$