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Quick continuity question

  1. Oct 24, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that x3 - 15x + 1 has three zeroes in the closed interval [-4,4].


    2. Relevant equations
    I think it's just subbing.


    3. The attempt at a solution

    Well, with that range, I think I can just list them as so:

    -4
    -3
    -2
    -1
    0
    1
    2
    3
    4

    From here, I figure out an answer by subbing each into each. And through the intermediate value theorem, if 2 numbers have a 0 between them, that should prove one zero is present. I continue this until I discover 3. So first, I when subbing in -4, I get:

    -64 + 50 + 1 = -3

    Subbing in -3 for x, we get:

    -27 + 45 + 1 = 19

    So there must be a 0 between -3 and 19. Thus, I have found one, I continue until I find 2 more?

    Am I doing this question right? Thanks for the help!
     
  2. jcsd
  3. Oct 24, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    Sure, if you can find three sign changes, then there are at least three roots.
     
  4. Oct 24, 2009 #3
    Thanks! I guess sign changes is a better way of looking at it. Appreciate the help!
     
  5. Oct 25, 2009 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Though the sign changes do not necessarily show up at integer values of x.

    For example, if the function is 0 at x= 1/2 and at x= 3/4, looking at x= 0 and 1 would miss that.
     
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