Analysis: Derivatives, Rolle's Theorem

[AlonsoMcLaren]

Homework Statement

If f has a finite third derivative f''' on (a,b) and if f(a)=f'(a)=f(b)=f'(b)=0 prove that f'''(c)=0 for some c in (a,b)

Homework Equations

Rolle's Theorem: Assume f has a derivative (finite or infinite) at each point of an open interval (a,b) and assume that f is continuous at both endpoints a and b. If f(a)=f(b) there is at least one interior point c at which f'(c)=0

The Attempt at a Solution

Because f is differentiable at a and b, f is continuous at a and b. Also, we have f(a)=f(b)=0

Therefore, use Rolle's theorem, there exists a<x<e such that f'(x)=0.

If we know that f' is continuous at a and b (I'M STUCK HERE), then we can conclude that there exists a<y<x such that f''(y)=0 and there exists x<z<b such that f''(z)=0. f''(y)=f''(z). Because f has a finite third derivative on (a,b) (If it is [a,b] then it's done...), f'' is continuous on (a,b) and therefore continuous at y and z. Therefore, by Rolle's theorem, there exists y<w<z such that f'''(w)=0..

So you know why I am stuck... Thank your for your help.

 

[superg33k]

My thoughts so far go along the lines of (but I don't really know what I'm talking about):
f(a)=f(b) so at least one f'(c)=0 on the closed interval [a,b].
But also f'(a)=f'(b)=0 so we have at least 3 turning points in f'(x) in (a,b).
Therefore need at least 2 turning points in f''(x) in (a,b)
Then at least 1 turning point in f'''(x) in (a,b), which is almost the answer.

 

[AlonsoMcLaren]

My thoughts so far go along the lines of (but I don't really know what I'm talking about):
f(a)=f(b) so at least one f'(c)=0 on the closed interval [a,b].
But also f'(a)=f'(b)=0 so we have at least 3 turning points in f'(x) in (a,b).
Therefore need at least 2 turning points in f''(x) in (a,b)
Then at least 1 turning point in f'''(x) in (a,b), which is almost the answer.

By which theorem?

 

[Dick]

By which theorem?

Mean Value Theorem. superg33k is saying you can show, as you have, that there is a c such that a<c<b such that f'(a)=f'(b)=f'(c)=0. What does that tell you about f'' on (a,c) and (c,b)? What does that tell you about f'''?

 

[AlonsoMcLaren]

What does that tell you about f'' on (a,c) and (c,b)?

It tells me nothing as f' is not guaranteed to be continuous at a and b so I cannot apply Mean Value Theorem or Rolle's Theorem. Please read my original post.

 

[Dick]

It tells me nothing as f' is not guaranteed to be continuous at a and b so I cannot apply Mean Value Theorem or Rolle's Theorem. Please read my original post.

If a function is differentiable at a point then it's automatically continuous at that point. You can prove that by looking at the difference quotient.

 

[AlonsoMcLaren]

If a function is differentiable at a point then it's automatically continuous at that point. You can prove that by looking at the difference quotient.

But we do not know that f' (NOT f) is differentiable at a and b.

 

[Dick]

But we do not know that f' (NOT f) is differentiable at a and b.

Ok, I see what you are saying. Finally. The only thing I can think of is to use Darboux's theorem. If f' is discontinuous at 0 then the discontinuity is essential, i.e. the limit x->0 f'(x) doesn't exist. That should make it hard for higher derivatives to be bounded. Seems a little much for a Rolle's theorem problem.