How to prove a smooth curve $c(t), t ∈ [a, b]$ is parameterized by arc length iff $a=0$ and $|c'(t)| = 1$.

Question

Here's the original problem:

Let $c = c(t)$ be a smooth (parametrized) curve in $\mathbb{R}^3$, $t ∈ [a, b]$. Show that $c$ is parametrized by arc length if and only if $a = 0$ and $|c'(t)| = 1$.