(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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)

2. Relevant 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

3. 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.

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# Homework Help: Analysis: Derivatives, Rolle's Theorem

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