- #1

boombaby

- 130

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## Homework Statement

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]

## 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 [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!