# Over what intervals is  f(x)=(9x^2-x)/(x-1)  increasing and decreasing?

Nov 16, 2015

$f$ is increasing on $\left(- \infty , \frac{3 - 2 \sqrt{2}}{3}\right)$ and on $\left(\frac{3 + 2 \sqrt{2}}{3} , \infty\right)$ and it is decreasing on $\left(\frac{3 - 2 \sqrt{2}}{3} , 1\right)$ and on $\left(1 , \frac{3 + 2 \sqrt{2}}{3}\right)$

#### Explanation:

$f ' \left(x\right) = \frac{\left(18 x - 1\right) \left(x - 1\right) - \left(9 {x}^{2} - x\right) \left(1\right)}{x - 1} ^ 2$

$= \frac{9 {x}^{2} - 18 x + 1}{x - 1} ^ 2$

$f ' \left(x\right) = 0$, at solutions to $9 {x}^{2} - 18 x + 1 = 0$.
Solve by completing the square of by using the quadratic formula.
$x = \frac{3 \pm 2 \sqrt{2}}{3}$

$f ' \left(x\right)$ is not defined at $x = 0$

Test the sign of $f '$ on each interval:

{: (bb"Interval:",(-oo,(3-2sqrt2)/3),((3-2sqrt2)/3,1),(1,(3+2sqrt2)/3),((3+2sqrt2)/3,oo)), (darrbb"Factors"darr,"========","======","=====","======"), (9x^2-18x+1, bb" +",bb" -",bb" -",bb" +"), ((x-1)^2,bb" +",bb" +",bb" +",bb" +"), ("==========","========","======","=====","======"), (bb"Product"=f'(x),bb" +",bb" -",bb" -",bb" +") :}