# Intervals of increase /decrease

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.

• You should already be acquainted with how to discuss the sign of a fraction, are you not? – Saucy O'Path Jun 8 '18 at 10:18
• Do you know the definition of a critical point? – Michael Burr Jun 8 '18 at 10:20

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 ?

• The equation that i wrote is already deviated... – Giulia Della Rosa Jun 8 '18 at 10:28
• It says that f′(x)>0 for x not being equal to -1 and f'(x) smaller than 0 when x is not equal to 1 – Giulia Della Rosa Jun 8 '18 at 10:29

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 $x=0$.