Extremes as Zeros

Let $f(x) = \dfrac{1}{x} - x + 2a\ln x$, where $a$ is a real number.

1. If $a = \dfrac{1}{2}$, find the equation of the tangent line to $y = f(x)$ at $(1, f(1))$. 2. Find the intervals on which $f(x)$ is monotonic, discussing all cases. 3. Let $g(x) = \ln x - bx - cx^2$. Suppose the two extreme points $x_1 < x_2$ of $f(x)$ are exactly the two zeros of $g(x)$, and the range of

is $\left[\ln 2 - \dfrac{2}{3}, +\infty\right)$. Find the range of possible values of $a$.