For example $f(x) = x^2$ is twice-differentiable. What special properties does $f(x) = x^2$ hold over a function that is differentiable but not twice-differentiable?
$\endgroup$ 31 Answer
$\begingroup$If you had one of those Lionel electric trains with an oval track when you were a kid, then you could see this in action. There were 10 pieces of track. 4 curved ones made a semi circle. Two semicircles were joined by two straight pieces. As you ran the train around the track (at full speed, of course) the forces would eventually pull the pieces of track apart.
There are 10 seams on the oval, but the track always came apart where the straight pieces abutted the curved pieces. Why? The oval was continuous and the tangent line was continuous. But at those 4 points where the straight met the curve, the the second derivative was discontinuous. There's a little jerk (pun intended) every time the train hit those points. It went from 0 centripetal force to something positive in 0 time.
People who design train tracks and roller coasters try to get several derivatives continuous in order to avoid such dangerous wear and tear at single points.
$\endgroup$ 1
