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.$