A Tangent of Slope $6$

Let $f(x)=\dfrac{1-\ln x}{x^2}$.

1. Find the zeros of $f(x)$ and its intervals of monotonicity. 2. Prove that the curve $y=\dfrac{\ln x}{x}$ has a tangent line of slope $6$, and that the $y$-coordinate $y_0$ of the point of tangency satisfies $y_0<-1$.