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Proving existence

  1. Sep 7, 2008 #1
    1. The problem statement, all variables and given/known data
    [tex]Let F(x) = (x-a)^2(x-b)^2 + x[/tex]. Show that the output [tex]\frac{a+b}{2}[/tex] exists for some value x.

    2. Relevant equations
    Quadratic formula. [tex]x^2 \geq 0[/tex].


    3. The attempt at a solution
    Hmm I've tried setting the two equal but that doesn't look nice (if I multiply everything out). It's easy to find the zeros of F(x) so there might be someway to relate to that? If someone could just give me a hint at a good first step for showing the existence of a certain output of a function.
     
  2. jcsd
  3. Sep 7, 2008 #2
    Actually, I just edited it since there was an x in there. Now if a = b, then the output (a+b)/2 has to exist right? I'm not sure how to "show" it though. Show is just a bit more informal than a proof right?
     
  4. Sep 7, 2008 #3

    Dick

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    F(a)=a and F(b)=b. That's a pretty good hint.
     
  5. Sep 7, 2008 #4
    So invoke the Intermediate Value Theorem?
     
  6. Sep 7, 2008 #5

    Dick

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    Exactly.
     
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