A function bounded and differentiable, but have an unbounded derivative?

[danielkyulee]

Can a function f: (a,b) in R be bounded and diffferentiablle, but have an unbounded derivative. I believe it can, but can not think of any examples where this is true. Anyone have any ideas?

 

[Kreizhn]

Sure, so long as you properly choose your domain.

So let's think about this, we want a function whose derivative approaches infinity, but such that the function itself is bounded. Probably the easiest way to do this is to find a function whose derivative goes to zero, then rotate about the axis y=x.

You don't need anything complicated. A monomial will do fine.

 

[Fredrik]

Try to think of a function with a graph that looks a bit like the graph of a periodic function, but is such that the distance between the peaks get shorter and shorter instead of staying the same.

 

[Kreizhn]

Try to think of a function with a graph that looks a bit like the graph of a periodic function, but is such that the distance between the peaks get shorter and shorter instead of staying the same.

Very nice. Perhaps somewhat more complicated, but nice.

 

[Ray Vickson]

Sqrt(x), 0 < x < 1.

 

[Fredrik]

Ray, here in the homework forum it's important to answer with hints, not complete solutions.