# Points of Inflection on a rational function

1. Aug 5, 2012

### Larsani

1. The problem statement, all variables and given/known data

Find the inflection points and use second derivative test to determine where the function is concave up or down

2. Relevant equations

f(x) = (x - 1)/(x2 - 4)

3. The attempt at a solution

f'(x) = (-x2 + 2x - 4) / (x2 - 4)2

f''(x) = 2x3 - 6x2 +24x -8 / (x2-4)4

This is where I am stuck. I can't solve the numerator set to 0. You can factor out the 2 and thats about it. Cubic that I cant solve. I'm looking at:

x(x2 - 3x +12) = 4

and thinking that maybe I can use the quadratic equation to find when that quadratic = 0 but I dont think that will help me.

2. Aug 5, 2012

### HallsofIvy

You want to solve $2x^3- 6x^2+ 24x- 8= 0$? The obvious first thing to do is to divide through by 2 to get $x^3- 3x^2+ 12x- 4= 0$. Now the only possible rational roots are $\pm 1, \pm 2$, and $\pm 4$. Trying those in turn, we see that none of them satisfy the equation so there is not any "simple" solution.