SOLUTION: Good day, I would like help with finding the intervals of increase and decrease of this function and its local minimum and maximum if it has them. Since it is a rational function I

Question 810983: Good day, I would like help with finding the intervals of increase and decrease of this function and its local minimum and maximum if it has them. Since it is a rational function I used the quotient rule to derive it but I don't think I'm getting it right. It is y = x/(x-1)^2.
Thank you.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
I see two ways to calculate the derivative, and I like that,
because I know I make mistakes, so if both answers agree I'm reassured,
and if they don not agree, I try to find and fix errors.
(I will not say how it went this time).
This is what I ended up with:

Using the quotient rule

The other way:
.
So is another expression for the function,
and it could be useful for derivative calculation and more.
Now I will calculate the derivative of

With both answers in agreement, now I am pretty sure that
is the derivative.
changes sign at and at .
For , where ,
so and the function is increasing at that point,
and at all points with .
The denominator of changes sign at ,
which we know is a vertical asymptote,
so changes sign at .
For and the function starts decreasing after , (but the function does not change sign there).
The numerator of changes sign at ,
so for and the function is decreasing.
Since the function decreases for ,
increases for ,
and for the function exists, with ,
the function has a local minimum at .
, and zooming in
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