One Common Point, Pinned Down

Let $f(x) = x\ln x + \dfrac{1}{x}$.

1. Find the intervals on which $f(x)$ is monotonic. 2. If the curve $y = f(x)$ and the line $y = kx - k$ (with $k > 0$) have exactly one point $P(x_0, y_0)$ in common, prove that $2 < x_0 < 3$.