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Let \(f\) be the function defined on \((0, +\infty)\) by:$$ f(x) = x \ln(x) $$
  1. Show that for all \(x \in (0, +\infty)\), \(f'(x) = \ln(x) + 1\).
  2. Study the sign of \(f'(x)\) and construct the table of variations for \(f\) on \((0, +\infty)\).

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