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**zebra2147** I came across this problem and need some guidance if anyone can help me...

Let $\displaystyle f: \mathbb{R}\rightarrow \mathbb{R}$ be such that $\displaystyle f '''$ exists. Suppose $\displaystyle f(a)=f(b)=f '(a)=f '(b)=0$ for some $\displaystyle a<b$.

1) Prove $\displaystyle f '''(c)=0$ for some $\displaystyle c$ contained in $\displaystyle (a,b)$.

2) If $\displaystyle f(x)= (x-a)^2(x-b)^2$ find a formula for $\displaystyle c$.

For part 1 I think it would be pretty easy to show that $\displaystyle f(c)=0$ and $\displaystyle f'(c)=0$ so I'm thinking once we know that we can use that. Also, I believe that $\displaystyle (f''(x)-f''(c))/(x-c)\rightarrow f'''(x) as x \rightarrow c$ which I'm guessing is going to be important to know.

Part 2, I'm pretty lost on even where to start. I would appreciate a push start.