Demonstrating that a function is monotonically increasing/decreasing

Question

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.

If $x_1<x_2<0$, then $-x_1>-x_2>0$ and hence $(-x_1)^2>(-x_2)^2>0$ or$x_1^2>x_2^2>0$. — Yiorgos S. Smyrlis, Oct 12 '14 at 14:06
Thanks.
Is it correct at the end to divide by $$x_{1}^{2}$$ and $$x_{2}^{2}$$ because this gives an inequality with the actual function? — jm22b, Oct 12 '14 at 14:14
Yes, it is correct, since it is as if you multiply both terms of the inequality by positive numbers. — Yiorgos S. Smyrlis, Oct 12 '14 at 14:16

score 3 · Accepted Answer · answered Oct 12 '14 at 14:08

3

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 $x>0$.

answered Oct 12 '14 at 14:08

emcor

792

We have just covered using the derivative for these types of questions in class, but in this case I needed to do it algebraically.
Thanks all the same though, I think I have it cracked.
– jm22b Oct 12 '14 at 14:15

score 2 · Answer 2 · answered Oct 12 '14 at 14:05

2

Hint: Just calculate $f'(x)$ see whether it is positive or negative or nonnegative or nonpositive in the domain .

answered Oct 12 '14 at 14:05

Ri-Li

9,038

And how does one check for a general continuous differentiable function $f(x)$ if it's derivative is, say, positive for all points on the interval $[a, b]$? I guess, one would need to find the roots of $f'(x)=0$ and check if they are outside this interval? But its not possible to find the roots in general, only numerically, no? – Confounded Jun 29 '21 at 23:22

score 2 · Answer 3 · answered Oct 12 '14 at 14:07

2

The answer lies in the derivative, $f'(x)=-\frac{2}{x^3}\gt 0,\,\forall x\lt 0$ thus the function is monotonic increasing.

answered Oct 12 '14 at 14:07

Ellya

11,783

Demonstrating that a function is monotonically increasing/decreasing

3 Answers3