How i can prove this function is not a monotone function without the derivative test? $$f(x)=-\frac{1}{x^3}$$ thanks in advance
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Welcome to Maths SX! It is monotonic on each interval that make up its domain. – Bernard Feb 06 '19 at 18:56
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It is monotone on any interval where it is defined. It is only not monotone when considered across the point where it is not defined ($x=0.$) – Thomas Andrews Feb 06 '19 at 18:57
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thanks sir i got it now – Feemo Fellow Feb 06 '19 at 19:05
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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: $x^3-y^3=(x-y)(x^2+xy+y^2).$
Kenta S
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