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Study \(f(x) = x \ln x - x\) on \(\mathbb{R}_+^*\):
  1. compute the limit of \(f\) at \(0^+\) (by continuity of the standard limits);
  2. determine the monotonicity intervals;
  3. study the convexity of \(f\) on \(\mathbb{R}_+^*\) and conclude on the existence (or not) of an inflection point.

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