Let $f$ be a differentiable function from $\mathbb{R}$ to $\mathbb{R}$ such that $|f(x)-f(y)|\le 2|x-y|^{3/2}$ for all $x,y\in\mathbb{R}$. If $f(0)=1$, then $\displaystyle \int_{0}^{1} f^{2}(x)\,dx$ is equal to: