In my introductory calculus textbook, I have just begun reading a section about finding the lengths of parametrically defined curves using integration. In the introduction to this section, my textbook makes the following assertion:
$C$ is a curve specified parametrically by the equation $x = f(t)$ and $y = g(t)$ for $a \le t \le b$. A smooth curve $C$ does not double back or reverse the direction of motion over the time interval $[a, b]$ since $(f')^2 + (g')^2 > 0$ throughout the interval. At a point where a curve does start to double back on itself, either the curve fails to be differentiable or both derivatives must simultaneously equal zero.
My understanding is that a smooth function is a function that has all order derivatives defined on its domain.
However, my textbook does not justify its assertion with any conceptual explanation or proof. I think this is a useful piece of information that would be enlightening and help me understand the concept much more effectively. As such, I would greatly appreciate it if people would be gracious enough to take the time to elaborate on this assertion and/or prove/show why this assertion is true.
For context, my calculus understanding is at a level of basic multivariable calculus.