## Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Let $$f:[0, \infty) \to [0,\infty)$$ be a differentiable function with $$f’$$ continuous. If $$f(f(x))=x^2$$, prove that $$\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$$ without explicitly finding $$f.$$

Since we are not allowed to determine $$f$$, I tried to apply the Cauchy Schwarz inequality for integrals in some ways, but it was not strong enough. I also noticed that $$f(x)=x^\sqrt{2}$$ verifies the given identity and I tried to get a lower bound for the integral from $$0 \leq \int_0^1 (f'(x)-\sqrt{2}x^{\sqrt{2}-1})^2dx$$ but this wasn’t good enough either.

I also managed to prove that $$f$$ is strictly increasing and $$f(0)=0, f(1)=1.$$