Question

Whats the largest number of points of inflection that can exist between two critical points of a continuous differentiable function?

Answer 1

Well, I know there HAS to be at least one. An inflection point is a point where a function changes curvature.

Answer 2

Let me delete the shame..

Answer 3

Well, sorry for all the dumb things I did.. was tired, I'll try to focus now =S
I have difficulties getting a graph for an example..
Let's think about it this way.. Can you think of a function that has any number you want of critical points between two points its value is 0?

Answer 4

I know it's indirect.. but still. could you? =]

Answer 5

Ok.. Seems like you're gone. I'll try to explain on my own.
Look at the following function:
http://www.wolframalpha.com/input/?i=x*sinx+%2B+30
It has many critical points between the first two points it has value of 0...

So let's define
\[
f'(x) = x \cdot \sin x + C
\]
the first 2 points f'(x) has value of 0 is depended on C, which could be as high as we want.

As you can see, f'(x) intends to be a derivative of some function f(x) (which we could find if we want.. doesn't matter really).
The curvature of f(x) is described by the rate of change of f'(x). if f'(x) goes bigger, it means f(x) gets bigger value faster and faster and therefore is convex at that point, and if f'(x) goes smaller then f(x) gets bigger less and less... and therefore is concave.

The rate of change, growing or shrinking, of f'(x) is described by f''(x)..
We know that a point with f''(x) = 0 is a point of inflection of f(x).
Without even calculating, we can tell that each critical point of our defined f'(x) has derivative 0 so f''(x) at that point is 0.

Since we can have any number of such critical points in f'(x) between two f'(x) = 0, that means we can have any number of inflection points on f(x) between two critical points of it.