= Solution
The domain is $x>0$, and $x\log x\to0$ from below as $x\downarrow0$. Its <derivative> is $1+\log x$, so it decreases up to $x=e^{-1}$ and then increases, with its unique global <minimum> at $(e^{-1},-e^{-1})$. It crosses the horizontal axis at $(1,0)$ and tends to $+\infty$ as $x\to\infty$. Its <second derivative> is $1/x>0$, so it is strictly <convex>. The right panel of the preceding figure shows these features; zero is an excluded endpoint with a finite limiting value, rather than a vertical <asymptote>.
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