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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?

  • Search for related questions, e.g. this one. Click on the linked questions on the left. – Dietrich Burde Sep 05 '17 at 20:24
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    You can take the second derivative of a twice-differentiable function. I know that seems like a too-obvious answer, but if you look around the Web for places where a twice-differentiable function is required, it often turns out there's a second derivative in a relevant formula that they want a function to satisfy. – David K Sep 05 '17 at 20:31
  • To answer the question in the title, the defining feature is of course that it's twice differentible... – Hans Lundmark Sep 06 '17 at 06:52

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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.

B. Goddard
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  • This is an excellent example which (I think?) hinges on Newton's Second Law relating force to acceleration as opposed to any other derivative of position. – benxyzzy Apr 09 '19 at 09:12