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Math Help - Intermedaite value theorm

  1. #1
    Member akhayoon's Avatar
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    Intermedaite value theorm

    given  f(x)=x^4+3x^2-x+1 show that there exist at least three real numbers x such that  f(x)=3

    how could I prove that there is at least three?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by akhayoon View Post
    given  f(x)=x^4+3x^2-x+1 show that there exist at least three real numbers x such that  f(x)=3

    how could I prove that there is at least three?
    this is not true. there are only two. check to make sure you typed the right problem
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  3. #3
    Member akhayoon's Avatar
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    rechecked confirmed, this is how the quiz question was written.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by akhayoon View Post
    rechecked confirmed, this is how the quiz question was written.
    Below are the graphs of y = x^4 + 3x^2 - x + 1 and y = 3. as you can clearly see, they intersect only twice. namely at x = -0.6337532429 and x = 0.8738358889. thus there is something wrong with the question.

    as you can see, the graph looks like a parabola, so, in fact, f(x) = c at most twice for any c
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Jhevon View Post
    this is not true. there are only two. check to make sure you typed the right problem
    I agree with Jhevon. (And so does my calculator. )

    -Dan
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  6. #6
    Member akhayoon's Avatar
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    ok, so would I use the I.V.T to prove them wrong?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by akhayoon View Post
    ok, so would I use the I.V.T to prove them wrong?
    the question does not have a "prove or disprove" instruction, so i would not really recommend it. just tell your professor that there is something wrong with the question.

    you can use IVT to prove there is at least 2. to disprove that there are at least 3 cannot be done with IVT as far as i can see, but we can disprove it (if that's allowed) using Rolle's theorem
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  8. #8
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    Quote Originally Posted by akhayoon View Post
    given  f(x)=x^4+3x^2-x+1 show that there exist at least three real numbers x such that  f(x)=3

    how could I prove that there is at least three?
    So the nearest correct problem might be

    Given f(x)=x^3+3x^2-x+1, then there exist at least three real numbers x such that f(x)=3 (See the graph below).
    Attached Thumbnails Attached Thumbnails Intermedaite value theorm-three-roots.gif  
    Last edited by curvature; December 8th 2007 at 09:53 PM.
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