Hi,

I am doing curve sketching in class and I don't understand an aspect of it. The curve to be sketched is $\displaystyle f(x)= \frac{x}{x-3} $

I understand the intercepts are (0,0) and (0,0), HA at y=1, VA at x=3 and domain all real except x=3 However, I am not sure how to find intervals of increase and decrease and relative extrema of the function

They have it listed as f'(x)= $\displaystyle \frac{-3}{(x-2)^2} < 0 $ concluding that f decreases everywhere and no local extemas --> how do they arrive at these conclusions?

Also they have no POI for the graph after taking f''(x)=$\displaystyle \frac{6}{(x-3)^2} $ How do they make this conclusion?