# f ''' and derivatives

• Jan 19th 2011, 09:57 PM
zebra2147
f ''' and derivatives
I came across this problem and need some guidance if anyone can help me...

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

1) Prove $f '''(c)=0$ for some $c$ contained in $(a,b)$.
2) If $f(x)= (x-a)^2(x-b)^2$ find a formula for $c$.

For part 1 I think it would be pretty easy to show that $f(c)=0$ and $f'(c)=0$ so I'm thinking once we know that we can use that. Also, I believe that $(f''(x)-f''(c))/(x-c)\rightarrow f'''(x) as x \rightarrow c$ 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.
• Jan 19th 2011, 10:04 PM
Drexel28
Quote:

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

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

1) Prove $f '''(c)=0$ for some $c$ contained in $(a,b)$.
2) If $f(x)= (x-a)^2(x-b)^2$ find a formula for $c$.

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

Hint:

Spoiler:

Rolle's theorem

Partial solution:

Spoiler:

For the first part I'd apply Rolle's theorem to find some $c_1\in(a,b)$ such that $f'(c_1)=0$. Note then that since $f'(a)=f'(c_1)=f'(b)=0$ there are points $c_2\in (a,c_1)$ and $c_3\in(c_2,b)$ such that $f''(c_2)=f''(c_3)=0$. Applying Rolle's theorem on that gives the result.

As for the second part it's just algebra?