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

1. Jul 20, 2011

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?

2. Jul 20, 2011

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.

3. Jul 20, 2011

Fredrik

Staff Emeritus
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.

4. Jul 20, 2011

Kreizhn

Very nice. Perhaps somewhat more complicated, but nice.

5. Jul 20, 2011

Ray Vickson

Sqrt(x), 0 < x < 1.

6. Jul 20, 2011

Fredrik

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