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.