# Homework Help: Inequality problem

1. Dec 8, 2009

### boombaby

1. The problem statement, all variables and given/known data
Let $$f\in C^{1}$$ on [0,1], and $$f(0)=f(1)=0$$, prove that
$$\int_{0}^{1}(f(x))^{2}dx \leq \frac{1}{4} \int_{0}^{1} (f'(x))^{2}dx$$

2. Relevant equations

3. 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!

2. Dec 9, 2009

### boombaby

anyone?........

3. Dec 9, 2009

### Billy Bob

Isn't this Wirtinger's Inequality?

4. Dec 9, 2009

5. Dec 9, 2009

### boombaby

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