my question is more of a conceptual one, but i'll use the problem i'm stuck on to keep things clear. I am confused about how to demonstrate whether a function is strictly monotonically increasing or decreasing etc. (i'm using the wrong brackets because the curly ones keep disappearing)
I have the function $$(f : x \in \mathbb{R} : x < 0) \rightarrow \mathbb{R}, f(x) = \frac{1}{x^{2}}$$
and I need to decide whether it is (strictly) monotonically increasing (or decreasing) and then show algebraically why this is the case.
I can see that it is strictly monotonically increasing and that it fits the inequality $$f(x_{1}) < f(x_{2})$$ for all $$x_{1}, x_{2} \in (-\infty, 0)$$ with $$x_{1} < x_{2}$$
but I am confused about how I show this algebraically. I'd really appreciate a general response to this that I can apply to similar problems.
Thank you very much.
Best Answer
You can see that your function is monotonically for $x<0$ increasing here:
http://www.wolframalpha.com/input/?i=1%2F%28x%5E2%29
For the left (right) side of $0$, you can show strictly increasing (decreasing) by the sign of the derivative:
$$f(x)'=\frac{-2}{x^3}$$
So $f$ is strictly increasing for $x<0$ and strictly decreasing for $x>0$.