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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 :)

Laszlo Stark
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  • I'd say it's fine how it is. Any briefer and you're likely to make it less clear. Compare with a computer program written in terse, codegolf-style language vs. longer but more natural language. You could write it with words ("is positive" rather than $>0$) or cases $\operatorname{sign}f'(x)=\begin{cases}+1\ldots\-1\end{cases}$, but I don't think that would add all that much. – Jam May 21 '23 at 08:15

2 Answers2

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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.)

Eike Schulte
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  • I guess you're right, I think I ended up with this mindset of trying to avoid text because I've become somewhat obsessed with concise notation but for the sake of clarity it's probably better to use text in certain cases. Also thanks for making me aware of the $\LaTeX$ notation :) – Laszlo Stark May 21 '23 at 08:22
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    Yeah, I think this obsession with concise notation is a natural phase to go through. However, your aim should be to communicate as clearly as possible, and text interspersed with formulas is usually the best way to achieve this (over purely textual or purely formulaic representations). – Eike Schulte May 21 '23 at 08:47
  • True about going through that phase, haha. Anyhow: f(x) is positive and negative, when x>2 and x<2, respectively is kinder than (x<2 ⟹ f′(x)<0) ∧ (x>2 ⟹ f′(x)>0). @LaszloStark – ryang May 21 '23 at 10:29
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I think what you have is probably fine. You could use "when" instead of for. You could use cases...

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

user317176
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  • Cases look really neat. I was actually writing "for" instead of "when" because my native language is german and I wasn't really sure which word to use, but you're probably right. Thank's for your help – Laszlo Stark May 21 '23 at 08:18
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    @LaszloStark "For" is more natural here for mathematical writing. – Jam May 21 '23 at 08:29