Math Help - Finding the range of...

1. Finding the range of...

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2. 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: $\left[\text{-}\tfrac{1}{10},\:\tfrac{1}{2}\right]$

3. Re: Finding the range of...

How would I find the range by complete the square?