Intervals of increase /decrease

Question

I would like to find the intervals on which the function $$\frac{-4x}{x^2-1},$$ increases and decrease, but I am not sure how? Could somebody help me? My textbook it says to first find the intervals and then the critical points. I have no clue how to do this, however.

Answer 1

If $f(x)=\frac{-4x}{x^2-1}$ , then $f'(x)=\frac{4+4x^2}{(x^2-1)^2}$

($x \ne 1$).

Hence $f'(x)>0$ for all $x \ne 1$.

Conclusion ?

Answer 2

You have to study the sign of the derivative: $$ f'(x)=\frac{-4x}{(x+1)(x-1)} $$ is positive when $x<-1$ or $0\le x < 1$, is negative when $-1< x \le 0$ or $x>1$, and is null when $x=0$.

So there is a local minimum in $x=0$.