Let $f(x)$ be a smooth function satisfying $$f(0)=f(2)=0$$ and $$\int_0^2 (f(x))^2 dx=1$$ and $$\int_0^2 (f'(x))^2 dx=1$$ Does such an f exist? Why? I'm (perhaps stupidly) presuming that this function doesn't exist but I can't intuitively think why, and I don't know how to go about the question mathematically. I know there must be a stationary point between 2 and 0 by definition. Any help would be greatly appreciated.