(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [tex]f\in C^{1}[/tex] on [0,1], and [tex]f(0)=f(1)=0[/tex], prove that

[tex]\int_{0}^{1}(f(x))^{2}dx \leq \frac{1}{4} \int_{0}^{1} (f'(x))^{2}dx[/tex]

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 [tex]f(x)=\int_{0}^{x} f'(t)dt[/tex] and f(1)=0 gives [tex]f(x)= -\int_{x}^{1} f'(t)dt[/tex]. But seems that I cannot go further.

Any small hint would be great, thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Inequality problem

**Physics Forums | Science Articles, Homework Help, Discussion**