I almost solve one problem, the thing is that to finish it I need to give an example of a function $f(x) \in C^1$ such that:
$$ \int_{1/4}^1 (f'(x))^2 dx \leq \int_{0}^{1/4} (f'(x))^2 dx $$
and $$f(0)=0, f(1)=1$$
My try: I was trying to construct a cycloid, that grow very fast before $\frac14$ and after it, that stay very flat until 1, but it doesn't work, also I tried, with the same idea like a $\log(x)$, it satisfies $(1)=1$ but I would like to "strech" that function in the begin so it satisfies the$ f(0)=0$. I would like something like this:
Can you find such function?
Take any function $f\in C^1$ such that $f(0)=0$ and $f(x)=1$ for all $x\in[1/4,1]$. For instance, $f(x)=8x(1-2x)$ if $x\in[0,1/4]$ and $f(x)=1$ if $x\in[1/4,1]$.