monotone function without derivative test

Question

How i can prove this function is not a monotone function without the derivative test? $$f(x)=-\frac{1}{x^3}$$ thanks in advance

Answer 1

This is not defined at $x=0$, so if you take an interval on which this function is defined, for example, $[a,b]$, either they are all positive or all negative. You can show that this function is monotone on any such interval by considering both cases, and using the equation: $x^3-y^3=(x-y)(x^2+xy+y^2).$