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Math Help - Polynomial of Degree 2

  1. #1
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    Polynomial of Degree 2

    My professor has this proof in his notes and I was wondering if anyone could help get me rolling...

    Let f :\mathbb{R} \rightarrow \mathbb{R} be differentiable. Suppose
    f(x + y) = f(x) + f(y) + 2xy for all x,y\in  \mathbb{R}.

    Show by completing the two parts below that f is a polynomial of degree 2.
    (1) Prove f'(x) = f'(0)+2x, for all x\in \mathbb{R}.
    (2) Prove f(x) = x^2+f'(0)x+f(0),for all x\in \mathbb{R}.

    I think my biggest problem is coming up with the best way to represent f'(x) which is obviously a big roadblock.
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  2. #2
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    Quote Originally Posted by zebra2147 View Post
    My professor has this proof in his notes and I was wondering if anyone could help get me rolling...

    Let f :\mathbb{R} \rightarrow \mathbb{R} be differentiable. Suppose
    f(x + y) = f(x) + f(y) + 2xy for all x,y\in  \mathbb{R}.

    Show by completing the two parts below that f is a polynomial of degree 2.
    (1) Prove f'(x) = f'(0)+2x, for all x\in \mathbb{R}.
    (2) Prove f(x) = x^2+f'(0)x+f(0),for all x\in \mathbb{R}.

    I think my biggest problem is coming up with the best way to represent f'(x) which is obviously a big roadblock.
    If you take the derivative with respect to x you get

    \displaystyle f'(x+y)=f'(x)+2y \implies f'(x)=f'(x+y)-2y

    This must hold for all y \in \mathbb{R} so what choice of y will give the result that you want?
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  3. #3
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    y=-x??

    So that we get f'(x) = f'(0)+2x?
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    Quote Originally Posted by zebra2147 View Post
    y=-x??

    So that we get f'(x) = f'(0)+2x?
    Yes that is the one
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    So is the information that you have helped me come up sufficient for the proof because it only has to work for all x and not all y?

    And do you have any hints for part 2?

    Thanks!
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  6. #6
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    Quote Originally Posted by zebra2147 View Post
    So is the information that you have helped me come up sufficient for the proof because it only has to work for all x and not all y?

    And do you have any hints for part 2?

    Thanks!
    Let g(x)=f(x)-x^2-f'(0)x. Note that g'(x)=f'(x)-2x-f'(0)=0 and since g'(x)=0 on an interval (in this case all of \mathbb{R}) we may conclude that f(x)-x^2-f'(0)x=g(x)=c for some c. Noting that g(0)=f(0) gives the desired results.
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