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Thread: f ''' and derivatives

  1. #1
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    f ''' and derivatives

    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.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zebra2147 View Post
    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.
    Hint:

    Spoiler:


    Rolle's theorem



    Partial solution:

    Spoiler:

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




    As for the second part it's just algebra?
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