I've heard that the monosyllabic adjectives – that is: adjectives which only consist of one syllable – take an umlaut in their comparative and superlative forms.
Is that true as a general rule?
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How do we know which adjective is monosyllabic in German?
A monosyllabic word has only one syllable (mono- → Greek for one or alone). Therefore, you will immediately notice if a word is monosyllabic or not, just by looking at it or reading it out.
Now let me comment on the claim that
"[...] monosyllabic adjectives take an umlaut in comparative and superlative [form]".
This claim is wrong.
Even the softened claim in @Janka's answer that this was true for "[m]ost common adjectives of one syllable whose stem- vowel is a, o, or u (not au)" is presumably untenable. Thinking about it for about ten minutes, I found only 12 adjectives for which it applies:
alt, arg, arm, dumm, groß, hoch, kalt, krank, kurz, lang, stark, warm, ...
There are certainly more; but yet, it was much easier to find the following (> 40) counter examples:
barsch, bunt, (cool), doll, (doof), drall, dumpf, fad, falsch, flach, froh, glatt, gut, harsch, hohl, hold, (hot), kahl, karg, klamm, klar, knapp, krumm, lasch, matt, oll, platt, prall, rasch, roh, rot, rund, satt, schroff, schlaff, schlank, schlapp, schmal, starr, stolz, straff, stumpf, stur, toll, voll, wahr, ...
The number of counter examples is so large that I would not consider these as mere exceptions from a common rule.
The often cited rule is actually more complicated:
However, as pointed out in the comments, that rule has a lot of exceptions as
Conclusion: Such a rule is pointless.
The key to understanding which words have vowel shifts and which not (includes i and e, too!) is vocabulary, not grammar. Short words are more likely to come from old German, which used vowel shifts for all sorts of derived forms. However, things have changed a lot over the last 1200 years so the only rule you can take from that is knowing your base vocabulary of short German words very well. All longer words are either made up from short ones or they follow simple rules, as they are later inventions.