Finding the range of...

• Oct 2nd 2012, 05:25 PM
Eraser147
Finding the range of...
Attachment 25015 Please click the image to enlarge.
• Oct 2nd 2012, 06:39 PM
Soroban
Re: Finding the range of...
Hello, Eraser147!

Quote:

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

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: $\left[\text{-}\tfrac{1}{10},\:\tfrac{1}{2}\right]$
• Oct 2nd 2012, 07:19 PM
Eraser147
Re: Finding the range of...
How would I find the range by complete the square?