Function recovered from a given arc length

Let $f:[0,\infty)\to[0,\infty)$ be a twice differentiable and increasing function with $f(0)=0$. Suppose that, for any $t\ge0$, the length of the arc of the curve $y=f(x)$, $x\ge0$ between $x=0$ and $x=t$ is $\frac23\left[(1+t)^{3/2}-1\right]$. Then $f(4)$ is equal to