find all open intervals on which the function f(x)= x/ x^2 + x -2 is decreasing
Let:
$\displaystyle f(x)=\frac{x}{x^2+2-2}$,
then:
$\displaystyle f'(x)=-\frac{x^2+2}{(x^2+x-2)^2}$
which is always $\displaystyle <0$ where it is defined. It is undefined at the roots of $\displaystyle x^2+x-2$ which are $\displaystyle x=-2$ and $\displaystyle x=1$.
Also $\displaystyle f(x)$ is decreasing where $\displaystyle f'(x)<0$.
Therefore $\displaystyle f(x)$ is decreasing on $\displaystyle (-\infty,-2),\ (-2,1) $ and $\displaystyle (1,\infty)$
RonL