# Thread: Increasing interval and decreasing interval

1. ## Increasing interval and decreasing interval

I have got a complex function y=(x^2-4*x-1)/(2*x+1) and I have to show the increasing and decreasing intervals.
How to show it when y'=(2*x^2+2*x-2)/(2*x+1)^2 but I am not able to find zero points and therefore it is impossible to find the increasing and decreasing intervals.

2. Originally Posted by totalnewbie
I have got a complex function y=(x^2-4*x-1)/(2*x+1) and I have to show the increasing and decreasing intervals.
How to show it when y'=(2*x^2+2*x-2)/(2*x+1)^2 but I am not able to find zero points and therefore it is impossible to find the increasing and decreasing intervals.
Sketch the curve:

3. This is not precise answer.

4. The function,
$y=\frac{x^2-4x-1}{2x+1},x\not =-1/2$ then,
$y'=\frac{2x^2+2x-2}{(2x+1)^2},x\not =-1/2$.
By Fermat's Principle the necessary conditions is when the derivative is zero or does not exits. Notice it does not exist at $x=-1/2$ but the function itself does not posses that domain. Thus, $y'=0$. That happens when $2x^2+2x-2=0$,
Thus,
$x^2+x-1=0$
$x=\frac{-1\pm\sqrt{5}}{2}$.
Now you can use derivative test to determine whether they are maximum or minimum or neither.

5. Originally Posted by totalnewbie
This is not precise answer.
No its not, its a hint.

You have two points at which the derivative is zero and one point at
which it goes to infinity which is between the zeros of the derivative.

It is increasing from -infinity to the first turning point decreasing
from there to the singularity, then decreasing from the singularity
to the next turning point and increasing from there out to +infinity.

RonL