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Math Help - Finding the range of...

  1. #1
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    Finding the range of...

    Finding the range of...-range.jpg Please click the image to enlarge.
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  2. #2
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    Re: Finding the range of...

    Hello, Eraser147!

    \text{Find the range of: }\:f(x) \:=\:\frac{x+2}{x^2+5}

    An informal sketch gives us a serpentine curve.
    Code:
                            |
                            |    *
                            | *  :  *
                            *    :    *
                  -5       *|    :        *
        -----------+------*-+----+-------------
          *        :     *  |    1
              *    :    *   |
                *  :  *     |
                   *        |
                            |
    The derivative simplifies to: . f'(x) \;=\;-\frac{(x-1)(x+5)}{(x^2+5)^2}

    Hence, the extrema are at: x = 1,\text{-}5

    And: . \begin{Bmatrix}f(1) &=& \frac{1}{2} \\ \\[-4mm] f(\text{-}5) &=& \text{-}\frac{1}{10} \end{Bmatrix}


    Therefore, the range is: \left[\text{-}\tfrac{1}{10},\:\tfrac{1}{2}\right]
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  3. #3
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    Re: Finding the range of...

    How would I find the range by complete the square?
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