Two Extreme Points and an Integer

Let $f(x) = x\ln x - \dfrac12 m x^2 - x$.

1. For $m = 4$, find the tangent line to $y = f(x)$ at $(1, f(1))$. 2. Suppose $f$ has two extreme points $x_1 < x_2$, and

holds for all admissible $m$. Find the largest integer $a$.