Inequality Problem: Prove f^2 ≤ 1/4 (f')^2

[boombaby]

Homework Statement

Let $$f\in C^{1}$$ on [0,1], and $$f(0)=f(1)=0$$, prove that
$$\int_{0}^{1}(f(x))^{2}dx \leq \frac{1}{4} \int_{0}^{1} (f'(x))^{2}dx$$

Homework Equations

The Attempt at a Solution

what is the trick to produce a 1/4 there? and how to make use of f(0)=f(1)=0? well I know that f(0)=0 gives a formula like $$f(x)=\int_{0}^{x} f'(t)dt$$ and f(1)=0 gives $$f(x)= -\int_{x}^{1} f'(t)dt$$. But seems that I cannot go further.
Any small hint would be great, thanks!

 

[boombaby]

anyone?...

 

[Billy Bob]

Isn't this Wirtinger's Inequality?

 

[Mark44]

There are some similarities. See http://en.wikipedia.org/wiki/Wirtinger's_inequality_for_functions

 

[boombaby]

yea, I'm reading it. cool. Thanks guys!