The problem is not a good problem, in the sense that what it asks you to prove is not true.
@ben_golub maybe that makes it a “good problem” for some skill testing? and by the way i agree x-> 1/x ln(x) is not monotonous.
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Not to mention that this problem makes more sense as a complex variables problem and the answer seems to assume real.
Here is a related question: which is bigger, e^Pi or Pi^e?
Answer is interesting: max x^y s.t. x*y=constant, and x=e no matter what the constant.
