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Math Help - Intermediate Value Theorem

  1. #1
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    Intermediate Value Theorem

    Hi!

    Considering the function

    f(x) =

    -x + x + 4 if x > -1
    4 if x = -1
    3 - x if x < -1

    I was asked to show that the equation f(x) = -1 has a solution between 1 and 2. (Function was already determined to be continuous.)

    For my solution:

    f(x) = -(1)^3 + 1 + 4 = 4
    f(x) = -(2)^3 + 2 + 4 = -2

    Given that f(x) is a continuous function and f(2) < -1 < f(1) then by the intermediate value theorem there exists some number c such that f(x) = -1

    Unfortuntely, I only recieved half credit for the answer. Is there something I missed? Was "f(2) < -1 < f(1)" the issue? Should I have manipulated my equations so that 0 would have been the center number?
    (ex. f(x) -x^3 + x + 4 + 1 = 0 )

    Any suggestions would be fantastic!
    Thank you!
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  2. #2
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    Quote Originally Posted by strgrl View Post
    Considering the function

    f(x) =
    -x + x + 4 if x > -1
    4 if x = -1
    3 - x if x < -1
    Please look over that definition. There must be a typo in it.
    Please correct it.
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  3. #3
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    Sorry!

    f(x) =
    -x^3 + x + 4 if x > -1
    4 if x = -1
    3 - x if x < -1

    Thank you.
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  4. #4
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    Quote Originally Posted by strgrl View Post
    I was asked to show that the equation f(x) = -1 has a solution between 1 and 2. (Function was already determined to be continuous.)
    Unfortuntely, I only recieved half credit for the answer. Is there something I missed? Was "f(2) < -1 < f(1)" the issue?
    Of course if I were you, I would indeed ask the instructor why.
    I can suggest two possible reasons (though I find neither reasonable here).
    1) Maybe you were expected to show f was continuous or at least note it.
    2) Maybe you were expected to find the actual value such f(c)=-1.
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