 Solution

The domain is $x>0$, and $x\log x\to 0$ from below as $x\downarrow 0$. 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.