Points of Inflection on a rational function

[Larsani]

Homework Statement

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

Homework Equations

f(x) = (x - 1)/(x² - 4)

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 that's about it. Cubic that I can't 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 don't think that will help me.

 

[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.