1. ## decreasing function

Find all open intervals on which the function $\displaystyle f(x) = \frac {x} {x^2 + x - 2}$ is decreasing.

2. f(x) = $\displaystyle \frac {x} {x^2+x-2}= \frac {x} {(x+2)(x-1)}$

$\displaystyle 1(x^2+x-2)-(2x+1)(x)$
$\displaystyle x^2+x-2-2x^2-x=-x^2-2$

f(prime)(x)= $\displaystyle \frac {-(x^2+2)}[\math] {((x+2)(x-1))^2$

3. Originally Posted by Samantha
Find all open intervals on which the function f(x) = \frac {x} {x^2 + x - 2} is decreasing.
$\displaystyle f^{\prime}(x) = \frac{1 \cdot (x^2 + x - 2) - x \cdot (2x + 1)}{(x^2 + x - 2)^2}$

$\displaystyle f^{\prime}(x) = \frac{-x^2 - 2}{(x^2 + x - 2)^2}$

When is this negative? (Hint: The denominator is always positive, since it is squared.)

-Dan

4. So it's always decreasing?

5. Originally Posted by Samantha
So it's always decreasing?
It looks like it.

-Dan

6. so it's (-infinity, -2), (-2,1) and (1, infinity)?

7. Originally Posted by Samantha
so it's (-infinity, -2), (-2,1) and (1, infinity)?
Yup. (You are more exact than I was going to be. I probably wouldn't have removed the points that are outside the domain, and you were right to do so. I would've lost points on this problem. )

-Dan