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Analysis Homework Help (2)

  1. Apr 20, 2009 #1
    1. The problem statement, all variables and given/known data
    Let f(x) = x^9+x^2+4. Prove: The equation f(x)=0 has at least one real solution.


    2. Relevant equations



    3. The attempt at a solution
    I know that the solution lies between -2 and -1. I also know that f(-2) = -504 and f(-1) = 4. I need to know how to use the IVT to prove that there is one real solution; not sure how to do that.
     
  2. jcsd
  3. Apr 20, 2009 #2

    dx

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    You've already done it. f is continuous so the intermediate value theorem applies. You showed that f(-1) = 4 and f(-2) is -504, and therefore there must be some c between -2 and -1 for which f(c) = 0 since 0 lies between 4 and -504.
     
  4. Apr 21, 2009 #3
    How would I write this up as a formal proof?
     
  5. Apr 21, 2009 #4

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    First you point out that f is continuous, and say that the IVT applies. Then you find f(-2) and f(-1), and say that 0 lies between these two values because one is positive and one is negative. Then, by the IVT, there must be some real number c between -2 and -1 such that f(c) = 0.
     
  6. Apr 21, 2009 #5
    Thanks so much for all your help!
     
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