find all open intervals on which the function f(x)= x/ x^2 + x -2 is decreasing

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- Jan 17th 2007, 10:22 PMgracyincreasing function
find all open intervals on which the function f(x)= x/ x^2 + x -2 is decreasing

- Jan 17th 2007, 10:47 PMCaptainBlack
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