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

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

    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?
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