Question

How do you find points of inflection of `(1-x^2)/x^3`?

Answer 1

It's probably best to start by writing this function as `f(x)=x^{-3}-x^{-1}`. Then `f'(x)=-3x^{-4}+x^{-2}` and `f''(x)=12x^{-5}-2x^{-3}=\frac{12-2x^{2}}{x^{5}}`. Setting `f''(x)=0` gives `x=\pm\sqrt{6}` as possible inflection points. These are actual inflection points because `f''(x)` changes sign as `x` increases through these values.