Finding the local extrema of $(1-x^2)/x^3$ without using calculus or a computer

Question

I am a student learning precalculus, and I have to solve the problem: find the local extrema of $\frac{1-x^2}{x^3}$. It is easy using a calculator or applying calculus, but I have no idea how to solve it without them, which is the way that our school is wanting. Can anyone help me? Thanks

Answer 1

Hints:

1. As an odd function, it is enough to study for $x>0$.
2. By AM-GM, $\dfrac1{x^3}+k^3+k^3 \geqslant 3k^2\dfrac1{x}$, now can you see what value of $k$ to select to get the minimum of $\dfrac1{x^3}-\dfrac1x$?