A function is such that: if is increasing, then ; and if is not increasing, then is not guaranteed. In fact, it is given that , and that is either increasing or decreasing. Which of the following must be true?

Answer and explanation

Correct answer: A. is decreasing

  1. Step 1: Let : 'increasing', : '', : 'decreasing'. We know , (since ) and .
  2. Step 2: From and we get . Then forces : is decreasing.
Answer
is decreasing.
Why not the others: 'Increasing' contradicts ; 'increasing and decreasing' also does; 'neither' contradicts ; '' contradicts the given fact.

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