Notate the value of a function f'(x) for a certain interval of x

Question

I'm currently working on a homework assignment where we have to describe certain characteristics of the graph of a function f. I'm trying to describe that the value of the derivative function f' is f'(x) < 0 for x < 2 and f'(x) > 0 for x > 2. I'm currently writing it just like that, using the word "for" two times, but I was wondering if there is a better way to notate this, substituting the word "for" with a symbol of some kind. (I'm currently learning LaTeX to do my homework and try to write as little text as possible in my maths notation as it makes for pretty bad readability in the document source)

I know this is probably really basic but I was struggeling to formulate this question concise enough such that a search engine might provide me with answers, hence I was hoping to find help on here.

Thanks in advance to everyone, I hope you have a great day :)

Answer 1

I think what you have is probably fine. You could use "when" instead of for. You could use cases...

$\begin{cases} f'(x) <0, & x<2\\f'(x) > 0, & x>2 \end{cases}$

Answer 2

Generally, you should not be trying to avoid text (and especially not just because the source code becomes harder to read). I would go with a sentence like:

For $x < 2$, we have $f'(x) < 0$ whereas $f'(x) > 0$ for $x > 2$.

The LaTeX code for this is

For $x < 2$, we have $f'(x) < 0$ whereas $f'(x) > 0$ for $x > 2$.

which I don’t think is bad.

(Also, you can use $\LaTeX$ notation on this site as well.)